Title:	Distributions for Generalized Additive Models for Location Scale and Shape
Version:	6.1-1
Date:	2023-08-01
Description:	A set of distributions which can be used for modelling the response variables in Generalized Additive Models for Location Scale and Shape, Rigby and Stasinopoulos (2005), <doi:10.1111/j.1467-9876.2005.00510.x>. The distributions can be continuous, discrete or mixed distributions. Extra distributions can be created, by transforming, any continuous distribution defined on the real line, to a distribution defined on ranges 0 to infinity or 0 to 1, by using a 'log' or a 'logit' transformation respectively.
License:	GPL-2 | GPL-3
URL:	https://www.gamlss.com/
BugReports:	https://github.com/mstasinopoulos/GAMLSS-Distibutions/issues
Depends:	R (≥ 3.5.0)
Imports:	MASS, graphics, stats, methods, grDevices
Suggests:	distributions3 (≥ 0.2.1)
Encoding:	UTF-8
NeedsCompilation:	yes
Packaged:	2023-08-23 19:47:28 UTC; dimitriosstasinopoulos
Author:	Mikis Stasinopoulos [aut, cre, cph], Robert Rigby [aut], Calliope Akantziliotou [ctb], Vlasios Voudouris [ctb], Gillian Heller [ctb], Fernanda De Bastiani [ctb], Raydonal Ospina [ctb], Nicoletta Motpan [ctb], Fiona McElduff [ctb], Majid Djennad [ctb], Marco Enea [ctb], Alexios Ghalanos [ctb], Christos Argyropoulos [ctb], Almond Stöcker [ctb], Jens Lichter [ctb], Stanislaus Stadlmann [ctb], Achim Zeileis [ctb]
Maintainer:	Mikis Stasinopoulos <d.stasinopoulos@gre.ac.uk>
Repository:	CRAN
Date/Publication:	2023-08-23 21:40:08 UTC

Distributions for Generalized Additive Models for Location Scale and Shape

Description

A set of distributions which can be used for modelling the response variables in Generalized Additive Models for Location Scale and Shape, Rigby and Stasinopoulos (2005), <doi:10.1111/j.1467-9876.2005.00510.x>. The distributions can be continuous, discrete or mixed distributions. Extra distributions can be created, by transforming, any continuous distribution defined on the real line, to a distribution defined on ranges 0 to infinity or 0 to 1, by using a 'log' or a 'logit' transformation respectively.

Details

The DESCRIPTION file:

Index of help topics:

BB                      Beta Binomial Distribution For Fitting a GAMLSS
                        Model
BCCG                    Box-Cox Cole and Green distribution (or Box-Cox
                        normal) for fitting a GAMLSS
BCPE                    Box-Cox Power Exponential distribution for
                        fitting a GAMLSS
BCT                     Box-Cox t distribution for fitting a GAMLSS
BE                      The beta distribution for fitting a GAMLSS
BEINF                   The beta inflated distribution for fitting a
                        GAMLSS
BEOI                    The one-inflated beta distribution for fitting
                        a GAMLSS
BEZI                    The zero-inflated beta distribution for fitting
                        a GAMLSS
BI                      Binomial distribution for fitting a GAMLSS
BNB                     Beta Negative Binomial distribution for fitting
                        a GAMLSS
DBI                     The Double binomial distribution
DBURR12                 The Discrete Burr type XII distribution for
                        fitting a GAMLSS model
DEL                     The Delaporte distribution for fitting a GAMLSS
                        model
DPO                     The Double Poisson distribution
EGB2                    The exponential generalized Beta type 2
                        distribution for fitting a GAMLSS
EXP                     Exponential distribution for fitting a GAMLSS
GA                      Gamma distribution for fitting a GAMLSS
GAF                     The Gamma distribution family
GAMLSS                  Create a GAMLSS Distribution
GB1                     The generalized Beta type 1 distribution for
                        fitting a GAMLSS
GB2                     The generalized Beta type 2 and generalized
                        Pareto distributions for fitting a GAMLSS
GEOM                    Geometric distribution for fitting a GAMLSS
                        model
GG                      Generalized Gamma distribution for fitting a
                        GAMLSS
GIG                     Generalized Inverse Gaussian distribution for
                        fitting a GAMLSS
GPO                     The generalised Poisson distribution
GT                      The generalized t distribution for fitting a
                        GAMLSS
GU                      The Gumbel distribution for fitting a GAMLSS
IG                      Inverse Gaussian distribution for fitting a
                        GAMLSS
IGAMMA                  Inverse Gamma distribution for fitting a GAMLSS
JSU                     The Johnson's Su distribution for fitting a
                        GAMLSS
JSUo                    The original Johnson's Su distribution for
                        fitting a GAMLSS
LG                      Logarithmic and zero adjusted logarithmic
                        distributions for fitting a GAMLSS model
LNO                     Log Normal distribution for fitting in GAMLSS
LO                      Logistic distribution for fitting a GAMLSS
LOGITNO                 Logit Normal distribution for fitting in GAMLSS
LQNO                    Normal distribution with a specific mean and
                        variance relationship for fitting a GAMLSS
                        model
MN3                     Multinomial distribution in GAMLSS
NBF                     Negative Binomial Family distribution for
                        fitting a GAMLSS
NBI                     Negative Binomial type I distribution for
                        fitting a GAMLSS
NBII                    Negative Binomial type II distribution for
                        fitting a GAMLSS
NET                     Normal Exponential t distribution (NET) for
                        fitting a GAMLSS
NO                      Normal distribution for fitting a GAMLSS
NO2                     Normal distribution (with variance as sigma
                        parameter) for fitting a GAMLSS
NOF                     Normal distribution family for fitting a GAMLSS
PARETO2                 Pareto distributions for fitting in GAMLSS
PE                      Power Exponential distribution for fitting a
                        GAMLSS
PIG                     The Poisson-inverse Gaussian distribution for
                        fitting a GAMLSS model
PO                      Poisson distribution for fitting a GAMLSS model
RG                      The Reverse Gumbel distribution for fitting a
                        GAMLSS
RGE                     Reverse generalized extreme family distribution
                        for fitting a GAMLSS
SEP                     The Skew Power exponential (SEP) distribution
                        for fitting a GAMLSS
SEP1                    The Skew exponential power type 1-4
                        distribution for fitting a GAMLSS
SHASH                   The Sinh-Arcsinh (SHASH) distribution for
                        fitting a GAMLSS
SI                      The Sichel dustribution for fitting a GAMLSS
                        model
SICHEL                  The Sichel distribution for fitting a GAMLSS
                        model
SIMPLEX                 The simplex distribution for fitting a GAMLSS
SN1                     Skew Normal Type 1 distribution for fitting a
                        GAMLSS
SN2                     Skew Normal Type 2 distribution for fitting a
                        GAMLSS
ST1                     The skew t distributions, type 1 to 5
TF                      t family distribution for fitting a GAMLSS
WARING                  Waring distribution for fitting a GAMLSS model
WEI                     Weibull distribution for fitting a GAMLSS
WEI2                    A specific parameterization of the Weibull
                        distribution for fitting a GAMLSS
WEI3                    A specific parameterization of the Weibull
                        distribution for fitting a GAMLSS
YULE                    Yule distribution for fitting a GAMLSS model
ZABB                    Zero inflated and zero adjusted Binomial
                        distribution for fitting in GAMLSS
ZABI                    Zero inflated and zero adjusted Binomial
                        distribution for fitting in GAMLSS
ZAGA                    The zero adjusted Gamma distribution for
                        fitting a GAMLSS model
ZAIG                    The zero adjusted Inverse Gaussian distribution
                        for fitting a GAMLSS model
ZANBI                   Zero inflated and zero adjusted negative
                        binomial distributions for fitting a GAMLSS
                        model
ZAP                     Zero adjusted poisson distribution for fitting
                        a GAMLSS model
ZIP                     Zero inflated poisson distribution for fitting
                        a GAMLSS model
ZIP2                    Zero inflated poisson distribution for fitting
                        a GAMLSS model
ZIPF                    The zipf and zero adjusted zipf distributions
                        for fitting a GAMLSS model
checklink               Set the Right Link Function for Specified
                        Parameter and Distribution
count_1_31              A set of functions to plot gamlss.family
                        distributions
exGAUS                  The ex-Gaussian distribution
flexDist                Non-parametric pdf from limited information
                        data
gamlss.dist-package     Distributions for Generalized Additive Models
                        for Location Scale and Shape
gamlss.family           Family Objects for fitting a GAMLSS model
gen.Family              Functions to generate log and logit
                        distributions from existing continuous
                        gamlss.family distributions
hazardFun               Hazard functions for gamlss.family
                        distributions
make.link.gamlss        Create a Link for GAMLSS families
momentSK                Sample and theoretical Moment and Centile
                        Skewness and Kurtosis Functions

Author(s)

NA

Maintainer: NA

References

Rigby, R. A. and Stasinopoulos D. M. (2005). Generalized additive models for location, scale and shape,(with discussion), Appl. Statist., 54, part 3, pp 507-554.

Rigby, R. A., Stasinopoulos, D. M., Heller, G. Z., and De Bastiani, F. (2019) Distributions for modeling location, scale, and shape: Using GAMLSS in R, Chapman and Hall/CRC, doi:10.1201/9780429298547. An older version can be found in https://www.gamlss.com/.

Stasinopoulos D. M. Rigby R.A. (2007) Generalized additive models for location scale and shape (GAMLSS) in R. Journal of Statistical Software, Vol. 23, Issue 7, Dec 2007, doi:10.18637/jss.v023.i07.

Stasinopoulos D. M., Rigby R.A., Heller G., Voudouris V., and De Bastiani F., (2017) Flexible Regression and Smoothing: Using GAMLSS in R, Chapman and Hall/CRC. doi:10.1201/b21973

(see also https://www.gamlss.com/).

See Also

gamlss.family

Examples

# pdf plot
plot(function(y) dSICHEL(y, mu=10, sigma = 0.1 , nu=1 ), 
              from=0, to=30, n=30+1, type="h")
# cdf plot
PPP <- par(mfrow=c(2,1))
plot(function(y) pSICHEL(y, mu=10, sigma =0.1, nu=1 ), 
             from=0, to=30, n=30+1, type="h") # cdf
cdf<-pSICHEL(0:30, mu=10, sigma=0.1, nu=1) 
sfun1  <- stepfun(1:30, cdf, f = 0)
plot(sfun1, xlim=c(0,30), main="cdf(x)")
par(PPP)

Beta Binomial Distribution For Fitting a GAMLSS Model

Description

This function defines the beta binomial distribution, a two parameter distribution, for a gamlss.family object to be used in a GAMLSS fitting using the function gamlss()

Usage

BB(mu.link = "logit", sigma.link = "log")
dBB(x, mu = 0.5, sigma = 1, bd = 10, log = FALSE)
pBB(q, mu = 0.5, sigma = 1, bd = 10, lower.tail = TRUE,
      log.p = FALSE)
qBB(p, mu = 0.5, sigma = 1, bd = 10, lower.tail = TRUE, 
       log.p = FALSE, fast = FALSE)
rBB(n, mu = 0.5, sigma = 1, bd = 10, fast = FALSE)

Arguments

Details

Definition file for beta binomial distribution.

f(y|\mu,\sigma)=\frac{\Gamma(n+1)} {\Gamma(y+1)\Gamma(n-y+1)} \frac{\Gamma(\frac{1}{\sigma}) \Gamma(y+\frac{\mu}{\sigma}) \Gamma[n+\frac{(1-\mu)}{\sigma}-y]}{\Gamma(n+\frac{1}{\sigma}) \Gamma(\frac{\mu}{\sigma}) \Gamma(\frac{1-\mu}{\sigma})}

for y=0,1,2,\ldots,n, 0<\mu<1 and \sigma>0, see pp. 523-524 of Rigby et al. (2019). . For \mu=0.5 and \sigma=0.5 the distribution is uniform.

Value

Returns a gamlss.family object which can be used to fit a Beta Binomial distribution in the gamlss() function.

Warning

The functions pBB and qBB are calculated using a laborious procedure so they are relatively slow.

Note

The response variable should be a matrix containing two columns, the first with the count of successes and the second with the count of failures. The parameter mu represents a probability parameter with limits 0 < \mu < 1. n \mu is the mean of the distribution where n is the binomial denominator. \{n \mu (1-\mu)[1+(n-1) \sigma/(\sigma+1)]\}^{0.5} is the standard deviation of the Beta Binomial distribution. Hence \sigma is a dispersion type parameter

Author(s)

Mikis Stasinopoulos, Bob Rigby and Kalliope Akantziliotou

References

Rigby, R. A. and Stasinopoulos D. M. (2005). Generalized additive models for location, scale and shape, (with discussion), Appl. Statist., 54, part 3, pp 507-554.

Rigby, R. A., Stasinopoulos, D. M., Heller, G. Z., and De Bastiani, F. (2019) Distributions for modeling location, scale, and shape: Using GAMLSS in R, Chapman and Hall/CRC, doi:10.1201/9780429298547. An older version can be found in https://www.gamlss.com/.

Stasinopoulos D. M. Rigby R.A. (2007) Generalized additive models for location scale and shape (GAMLSS) in R. Journal of Statistical Software, Vol. 23, Issue 7, Dec 2007, doi:10.18637/jss.v023.i07.

Stasinopoulos D. M., Rigby R.A., Heller G., Voudouris V., and De Bastiani F., (2017) Flexible Regression and Smoothing: Using GAMLSS in R, Chapman and Hall/CRC. doi:10.1201/b21973

(see also https://www.gamlss.com/).

See Also

gamlss.family, BI,

Examples

# BB()# gives information about the default links for the Beta Binomial distribution 
#plot the pdf
plot(function(y) dBB(y, mu = .5, sigma = 1, bd =40), from=0, to=40, n=40+1, type="h")
#calculate the cdf and plotting it
ppBB <- pBB(seq(from=0, to=40), mu=.2 , sigma=3, bd=40)
plot(0:40,ppBB, type="h")
#calculating quantiles and plotting them  
qqBB <- qBB(ppBB, mu=.2 , sigma=3, bd=40)
plot(qqBB~ ppBB)
# when the argument fast is useful
p <- pBB(c(0,1,2,3,4,5), mu=.01 , sigma=1, bd=5)
qBB(p, mu=.01 , sigma=1, bd=5, fast=TRUE)
#  0 1 1 2 3 5
qBB(p, mu=.01 , sigma=1, bd=5, fast=FALSE)
#  0 1 2 3 4 5
# generate random sample
tN <- table(Ni <- rBB(1000, mu=.2, sigma=1, bd=20))
r <- barplot(tN, col='lightblue')
# fitting a model 
# library(gamlss)
#data(aep)   
# fits a Beta-Binomial model
#h<-gamlss(y~ward+loglos+year, sigma.formula=~year+ward, family=BB, data=aep) 

Box-Cox Cole and Green distribution (or Box-Cox normal) for fitting a GAMLSS

Description

The function BCCG defines the Box-Cox Cole and Green distribution (Box-Cox normal), a three parameter distribution, for a gamlss.family object to be used in GAMLSS fitting using the function gamlss(). The functions dBCCG, pBCCG, qBCCG and rBCCG define the density, distribution function, quantile function and random generation for the specific parameterization of the Box-Cox Cole and Green distribution. [The function BCCGuntr() is the original version of the function suitable only for the untruncated Box-Cox Cole and Green distribution See Cole and Green (1992) and Rigby and Stasinopoulos (2003a, 2003b) for details. The function BCCGo is identical to BCCG but with log link for mu.

Usage

BCCG(mu.link = "identity", sigma.link = "log", nu.link = "identity")
BCCGo(mu.link = "log", sigma.link = "log", nu.link = "identity")
BCCGuntr(mu.link = "identity", sigma.link = "log", nu.link = "identity")
dBCCG(x, mu = 1, sigma = 0.1, nu = 1, log = FALSE)
pBCCG(q, mu = 1, sigma = 0.1, nu = 1, lower.tail = TRUE, log.p = FALSE)
qBCCG(p, mu = 1, sigma = 0.1, nu = 1, lower.tail = TRUE, log.p = FALSE)
rBCCG(n, mu = 1, sigma = 0.1, nu = 1)
dBCCGo(x, mu = 1, sigma = 0.1, nu = 1, log = FALSE)
pBCCGo(q, mu = 1, sigma = 0.1, nu = 1, lower.tail = TRUE, log.p = FALSE)
qBCCGo(p, mu = 1, sigma = 0.1, nu = 1, lower.tail = TRUE, log.p = FALSE)
rBCCGo(n, mu = 1, sigma = 0.1, nu = 1)

Arguments

Details

The probability distribution function of the untrucated Box-Cox Cole and Green distribution, BCCGuntr, is defined as

f(y|\mu,\sigma,\nu)=\frac{1}{\sqrt{2\pi}\sigma}\frac{y^{\nu-1}}{\mu^\nu} \exp(-\frac{z^2}{2})

where if \nu \neq 0 then z=[(y/\mu)^{\nu}-1]/(\nu \sigma) else z=\log(y/\mu)/\sigma, for y>0, \mu>0, \sigma>0 and \nu=(-\infty,+\infty).

The Box-Cox Cole and Green distribution, BCCG, adjusts the above density f(y|\mu,\sigma,\nu) for the truncation resulting from the condition y>0. See Rigby and Stasinopoulos (2003a, 2003b) or pp. 439-441 of Rigby et al. (2019) for details.

Value

BCCG() returns a gamlss.family object which can be used to fit a Cole and Green distribution in the gamlss() function. dBCCG() gives the density, pBCCG() gives the distribution function, qBCCG() gives the quantile function, and rBCCG() generates random deviates.

Warning

The BCCGuntr distribution may be unsuitable for some combinations of the parameters (mainly for large \sigma) where the integrating constant is less than 0.99. A warning will be given if this is the case. The BCCG distribution is suitable for all combinations of the distributional parameters within their range [i.e. \mu>0, \sigma>0, \nu=(-\infty, +\infty)]

Note

\mu is the median of the distribution \sigma is approximately the coefficient of variation (for small values of \sigma), and \nu controls the skewness.

The BCCG distribution is suitable for all combinations of the parameters within their ranges [i.e. \mu>0,\sigma>0, {\rm and} \nu=(-\infty,\infty) ]

Author(s)

Mikis Stasinopoulos, Bob Rigby and Kalliope Akantziliotou

References

Cole, T. J. and Green, P. J. (1992) Smoothing reference centile curves: the LMS method and penalized likelihood, Statist. Med. 11, 1305–1319

Rigby, R. A. and Stasinopoulos, D. M. (2004). Smooth centile curves for skew and kurtotic data modelled using the Box-Cox Power Exponential distribution. Statistics in Medicine, 23: 3053-3076.

Rigby, R. A. and Stasinopoulos D. M. (2005). Generalized additive models for location, scale and shape,(with discussion), Appl. Statist., 54, part 3, pp 507-554.

Rigby, R.A. Stasinopoulos, D.M. (2006). Using the Box-Cox t distribution in GAMLSS to mode skewnees and and kurtosis. Statistical Modelling, 6(3) :209. doi:10.1191/1471082X06st122oa

Rigby, R. A., Stasinopoulos, D. M., Heller, G. Z., and De Bastiani, F. (2019) Distributions for modeling location, scale, and shape: Using GAMLSS in R, Chapman and Hall/CRC. doi:10.1201/9780429298547 An older version can be found in https://www.gamlss.com/.

Stasinopoulos D. M. Rigby R.A. (2007) Generalized additive models for location scale and shape (GAMLSS) in R. Journal of Statistical Software, Vol. 23, Issue 7, Dec 2007, doi:10.18637/jss.v023.i07.

Stasinopoulos D. M., Rigby R.A., Heller G., Voudouris V., and De Bastiani F., (2017) Flexible Regression and Smoothing: Using GAMLSS in R, Chapman and Hall/CRC. doi:10.1201/b21973

(see also https://www.gamlss.com/).

See Also

gamlss.family, BCPE, BCT

Examples

BCCG()   # gives information about the default links for the Cole and Green distribution 
# library(gamlss)
#data(abdom)
#h<-gamlss(y~cs(x,df=3), sigma.formula=~cs(x,1), family=BCCG, data=abdom) 
#plot(h)
plot(function(x) dBCCG(x, mu=5,sigma=.5,nu=-1), 0.0, 20, 
 main = "The BCCG  density mu=5,sigma=.5,nu=-1")
plot(function(x) pBCCG(x, mu=5,sigma=.5,nu=-1), 0.0, 20, 
 main = "The BCCG  cdf mu=5, sigma=.5, nu=-1")

Box-Cox Power Exponential distribution for fitting a GAMLSS

Description

This function defines the Box-Cox Power Exponential distribution, a four parameter distribution, for a gamlss.family object to be used for a GAMLSS fitting using the function gamlss().

The functions dBCPE, pBCPE, qBCPE and rBCPE define the density, distribution function, quantile function and random generation for the Box-Cox Power Exponential distribution.

The function checkBCPE (very old) can be used, typically when a BCPE model is fitted, to check whether there exit a turning point of the distribution close to zero. It give the number of values of the response below their minimum turning point and also the maximum probability of the lower tail below minimum turning point.

[The function Biventer() is the original version of the function suitable only for the untruncated BCPE distribution.] See Rigby and Stasinopoulos (2003) for details.

The function BCPEo is identical to BCPE but with log link for mu.

Usage

BCPE(mu.link = "identity", sigma.link = "log", nu.link = "identity",
          tau.link = "log")
BCPEo(mu.link = "log", sigma.link = "log", nu.link = "identity", 
          tau.link = "log")
BCPEuntr(mu.link = "identity", sigma.link = "log", nu.link = "identity", 
          tau.link = "log")
dBCPE(x, mu = 5, sigma = 0.1, nu = 1, tau = 2, log = FALSE)
pBCPE(q, mu = 5, sigma = 0.1, nu = 1, tau = 2, lower.tail = TRUE, log.p = FALSE)
qBCPE(p, mu = 5, sigma = 0.1, nu = 1, tau = 2, lower.tail = TRUE, log.p = FALSE)          
rBCPE(n, mu = 5, sigma = 0.1, nu = 1, tau = 2)
dBCPEo(x, mu = 5, sigma = 0.1, nu = 1, tau = 2, log = FALSE)
pBCPEo(q, mu = 5, sigma = 0.1, nu = 1, tau = 2, lower.tail = TRUE, 
       log.p = FALSE)
qBCPEo(p, mu = 5, sigma = 0.1, nu = 1, tau = 2, lower.tail = TRUE, 
       log.p = FALSE)          
rBCPEo(n, mu = 5, sigma = 0.1, nu = 1, tau = 2)
checkBCPE(obj = NULL, mu = 10, sigma = 0.1, nu = 0.5, tau = 2,...)

Arguments

Details

The probability density function of the untrucated Box Cox Power Exponential distribution, (BCPE.untr), is defined as

f(y|\mu,\sigma,\nu,\tau)=\frac{y^{\nu-1} \tau \exp[-\frac{1}{2}|\frac{z}{c}|^\tau]}{\mu^{\nu} \sigma c 2^{(1+1/\tau)} \Gamma(\frac{1}{\tau})}

where c = [ 2^{(-2/\tau)}\Gamma(1/\tau)/\Gamma(3/\tau)]^{0.5}, where if \nu \neq 0 then z=[(y/\mu)^{\nu}-1]/(\nu \sigma) else z=\log(y/\mu)/\sigma, for y>0, \mu>0, \sigma>0, \nu=(-\infty,+\infty) and \tau>0 see pp. 450-451 of Rigby et al. (2019).

The Box-Cox Power Exponential, BCPE, adjusts the above density f(y|\mu,\sigma,\nu,\tau) for the truncation resulting from the condition y>0. See Rigby and Stasinopoulos (2003) for details.

Value

BCPE() returns a gamlss.family object which can be used to fit a Box Cox Power Exponential distribution in the gamlss() function. dBCPE() gives the density, pBCPE() gives the distribution function, qBCPE() gives the quantile function, and rBCPE() generates random deviates.

Warning

The BCPE.untr distribution may be unsuitable for some combinations of the parameters (mainly for large \sigma) where the integrating constant is less than 0.99. A warning will be given if this is the case.

The BCPE distribution is suitable for all combinations of the parameters within their ranges [i.e. \mu>0,\sigma>0, \nu=(-\infty,\infty) {\rm and} \tau>0 ]

Note

\mu, is the median of the distribution, \sigma is approximately the coefficient of variation (for small \sigma and moderate nu>0), \nu controls the skewness and \tau the kurtosis of the distribution

Author(s)

Mikis Stasinopoulos, Bob Rigby and Calliope Akantziliotou

References

Rigby, R. A. and Stasinopoulos D. M. (2005). Generalized additive models for location, scale and shape,(with discussion), Appl. Statist., 54, part 3, pp 507-554.

Rigby, R. A. and Stasinopoulos, D. M. (2004). Smooth centile curves for skew and kurtotic data modelled using the Box-Cox Power Exponential distribution. Statistics in Medicine, 23: 3053-3076.

Rigby, R. A., Stasinopoulos, D. M., Heller, G. Z., and De Bastiani, F. (2019) Distributions for modeling location, scale, and shape: Using GAMLSS in R, Chapman and Hall/CRC, doi:10.1201/9780429298547. An older version can be found in https://www.gamlss.com/.

Stasinopoulos D. M. Rigby R.A. (2007) Generalized additive models for location scale and shape (GAMLSS) in R. Journal of Statistical Software, Vol. 23, Issue 7, Dec 2007, doi:10.18637/jss.v023.i07.

Stasinopoulos D. M., Rigby R.A., Heller G., Voudouris V., and De Bastiani F., (2017) Flexible Regression and Smoothing: Using GAMLSS in R, Chapman and Hall/CRC. doi:10.1201/b21973

(see also https://www.gamlss.com/).

See Also

gamlss.family, BCT

Examples

# BCPE()   #
# library(gamlss) 
# data(abdom)
#h<-gamlss(y~cs(x,df=3), sigma.formula=~cs(x,1), family=BCPE, data=abdom) 
#plot(h)
plot(function(x)dBCPE(x, mu=5,sigma=.5,nu=1, tau=3), 0.0, 15, 
 main = "The BCPE  density mu=5,sigma=.5,nu=1, tau=3")
plot(function(x) pBCPE(x, mu=5,sigma=.5,nu=1, tau=3), 0.0, 15, 
 main = "The BCPE  cdf mu=5, sigma=.5, nu=1, tau=3")

Box-Cox t distribution for fitting a GAMLSS

Description

The function BCT() defines the Box-Cox t distribution, a four parameter distribution, for a gamlss.family object to be used in GAMLSS fitting using the function gamlss().

The functions dBCT, pBCT, qBCT and rBCT define the density, distribution function, quantile function and random generation for the Box-Cox t distribution.

[The function BCTuntr() is the original version of the function suitable only for the untruncated BCT distribution]. See Rigby and Stasinopoulos (2003) for details.

The function BCTo is identical to BCT but with log link for mu.

Usage

BCT(mu.link = "identity", sigma.link = "log", nu.link = "identity", 
          tau.link = "log")
BCTo(mu.link = "log", sigma.link = "log", nu.link = "identity", 
          tau.link = "log")
BCTuntr(mu.link = "identity", sigma.link = "log", nu.link = "identity", 
          tau.link = "log")
dBCT(x, mu = 5, sigma = 0.1, nu = 1, tau = 2, log = FALSE)
pBCT(q, mu = 5, sigma = 0.1, nu = 1, tau = 2, lower.tail = TRUE, log.p = FALSE)
qBCT(p, mu = 5, sigma = 0.1, nu = 1, tau = 2, lower.tail = TRUE, log.p = FALSE)          
rBCT(n, mu = 5, sigma = 0.1, nu = 1, tau = 2)
dBCTo(x, mu = 5, sigma = 0.1, nu = 1, tau = 2, log = FALSE)
pBCTo(q, mu = 5, sigma = 0.1, nu = 1, tau = 2, lower.tail = TRUE, log.p = FALSE)
qBCTo(p, mu = 5, sigma = 0.1, nu = 1, tau = 2, lower.tail = TRUE, log.p = FALSE)          
rBCTo(n, mu = 5, sigma = 0.1, nu = 1, tau = 2)

Arguments

Details

The probability density function of the untruncated Box-Cox t distribution, BCTuntr, is given by

f(y|\mu,\sigma,\nu,\tau)=\frac{y^{\nu-1}}{\mu^{\nu}\sigma} \frac{\Gamma[(\tau+1)/2]}{\Gamma(1/2) \Gamma(\tau/2) \tau^{0.5}} [1+(1/\tau)z^2]^{-(\tau+1)/2}

where if \nu \neq 0 then z=[(y/\mu)^{\nu}-1]/(\nu \sigma) else z=\log(y/\mu)/\sigma, for y>0, \mu>0, \sigma>0, \nu=(-\infty,+\infty) and \tau>0 see pp. 450-451 of Rigby et al. (2019).

The Box-Cox t distribution, BCT, adjusts the above density f(y|\mu,\sigma,\nu,\tau) for the truncation resulting from the condition y>0. See Rigby and Stasinopoulos (2003) for details.

Value

BCT() returns a gamlss.family object which can be used to fit a Box Cox-t distribution in the gamlss() function. dBCT() gives the density, pBCT() gives the distribution function, qBCT() gives the quantile function, and rBCT() generates random deviates.

Warning

The use BCTuntr distribution may be unsuitable for some combinations of the parameters (mainly for large \sigma) where the integrating constant is less than 0.99. A warning will be given if this is the case.

The BCT distribution is suitable for all combinations of the parameters within their ranges [i.e. \mu>0,\sigma>0, \nu=(-\infty,\infty) {\rm and} \tau>0 ]

Note

\mu is the median of the distribution, \sigma(\frac{\tau}{\tau-2})^{0.5} is approximate the coefficient of variation (for small \sigma and moderate nu>0 and moderate or large \tau), \nu controls the skewness and \tau the kurtosis of the distribution

Author(s)

Mikis Stasinopoulos, Bob Rigby and Calliope Akantziliotou

References

Rigby, R. A. and Stasinopoulos D. M. (2005). Generalized additive models for location, scale and shape,(with discussion), Appl. Statist., 54, part 3, pp 507-554.

Rigby, R.A. Stasinopoulos, D.M. (2006). Using the Box-Cox t distribution in GAMLSS to mode skewnees and and kurtosis. Statistical Modelling 6(3):200. doi:10.1191/1471082X06st122oa.

Rigby, R. A., Stasinopoulos, D. M., Heller, G. Z., and De Bastiani, F. (2019) Distributions for modeling location, scale, and shape: Using GAMLSS in R, Chapman and Hall/CRC, doi:10.1201/9780429298547. An older version can be found in https://www.gamlss.com/.

Stasinopoulos D. M. Rigby R.A. (2007) Generalized additive models for location scale and shape (GAMLSS) in R. Journal of Statistical Software, Vol. 23, Issue 7, Dec 2007, doi:10.18637/jss.v023.i07.

Stasinopoulos D. M., Rigby R.A., Heller G., Voudouris V., and De Bastiani F., (2017) Flexible Regression and Smoothing: Using GAMLSS in R, Chapman and Hall/CRC. doi:10.1201/b21973

(see also https://www.gamlss.com/).

See Also

gamlss.family, BCPE, BCCG

Examples

BCT()   # gives information about the default links for the Box Cox t distribution
# library(gamlss)
#data(abdom)
#h<-gamlss(y~cs(x,df=3), sigma.formula=~cs(x,1), family=BCT, data=abdom) # 
#plot(h)
plot(function(x)dBCT(x, mu=5,sigma=.5,nu=1, tau=2), 0.0, 20, 
 main = "The BCT  density mu=5,sigma=.5,nu=1, tau=2")
plot(function(x) pBCT(x, mu=5,sigma=.5,nu=1, tau=2), 0.0, 20, 
 main = "The BCT  cdf mu=5, sigma=.5, nu=1, tau=2")

The beta distribution for fitting a GAMLSS

Description

The functions BE() and BEo() define the beta distribution, a two parameter distribution, for a gamlss.family object to be used in GAMLSS fitting using the function gamlss(). BE() has mean equal to the parameter mu and sigma as scale parameter, see below. BEo() is the original parameterizations of the beta distribution as in dbeta() with shape1=mu and shape2=sigma. The functions dBE and dBEo, pBE and pBEo, qBE and qBEo and finally rBE and rBE define the density, distribution function, quantile function and random generation for the BE and BEo parameterizations respectively of the beta distribution.

Usage

BE(mu.link = "logit", sigma.link = "logit")
dBE(x, mu = 0.5, sigma = 0.2, log = FALSE)
pBE(q, mu = 0.5, sigma = 0.2, lower.tail = TRUE, log.p = FALSE)
qBE(p, mu = 0.5, sigma = 0.2, lower.tail = TRUE, log.p = FALSE)
rBE(n, mu = 0.5, sigma = 0.2)
BEo(mu.link = "log", sigma.link = "log")
dBEo(x, mu = 0.5, sigma = 0.2, log = FALSE)
pBEo(q, mu = 0.5, sigma = 0.2, lower.tail = TRUE, log.p = FALSE)
qBEo(p, mu = 0.5, sigma = 0.2, lower.tail = TRUE, log.p = FALSE)

Arguments

Details

The standard parametrization of the beta distribution is given as:

f(y|\alpha,\beta)=\frac{1}{B(\alpha, \beta)} y^{\alpha-1}(1-y)^{\beta-1}

for y=(0,1), \alpha>0 and \beta>0.

The first gamlss implementation the beta distribution is called BEo, and it is identical to the standard parametrization with \alpha=\mu and \beta = \sigma, see pp. 460-461 of Rigby et al. (2019):

f(y|\mu,\sigma)=\frac{1}{B(\mu, \sigma)} y^{\mu-1}(1-y)^{\sigma-1}

for y=(0,1), \mu>0 and \sigma>0. The problem with this parametrization is that with mean E(y)=\mu/(\mu+\sigma) it is not convenient for modelling the response y as function of the explanatory variables. The second parametrization, BE see pp. 461-463 of Rigby et al. (2019), is using

\mu=\frac{\alpha}{\alpha+\beta}

\sigma= \frac{1}{\alpha+\beta+1)^{1/2}}

(of the standard parametrization) and it is more convenient because \mu now is the mean with variance equal to Var(y)=\sigma^2 \mu (1-\mu). Note however that 0<\mu<1 and 0<\sigma<1.

Value

BE() and BEo() return a gamlss.family object which can be used to fit a beta distribution in the gamlss() function.

Note

Note that for BE, mu is the mean and sigma a scale parameter contributing to the variance of y

Author(s)

Bob Rigby and Mikis Stasinopoulos

References

Rigby, R. A. and Stasinopoulos D. M. (2005). Generalized additive models for location, scale and shape,(with discussion), Appl. Statist., 54, part 3, pp 507-554.

Rigby, R. A., Stasinopoulos, D. M., Heller, G. Z., and De Bastiani, F. (2019) Distributions for modeling location, scale, and shape: Using GAMLSS in R, Chapman and Hall/CRC, doi:10.1201/9780429298547. An older version can be found in https://www.gamlss.com/.

Stasinopoulos D. M., Rigby R.A. and Akantziliotou C. (2006) Instructions on how to use the GAMLSS package in R. Accompanying documentation in the current GAMLSS help files, (see also https://www.gamlss.com/).

Stasinopoulos D. M. Rigby R.A. (2007) Generalized additive models for location scale and shape (GAMLSS) in R. Journal of Statistical Software, Vol. 23, Issue 7, Dec 2007, doi:10.18637/jss.v023.i07.

Stasinopoulos D. M., Rigby R.A., Heller G., Voudouris V., and De Bastiani F., (2017) Flexible Regression and Smoothing: Using GAMLSS in R, Chapman and Hall/CRC. doi:10.1201/b21973

(see also https://www.gamlss.com/).

See Also

gamlss.family, BE, LOGITNO, GB1, BEINF

Examples

BE()# gives information about the default links for the beta distribution
dat1<-rBE(100, mu=.3, sigma=.5)
hist(dat1)        
#library(gamlss)
# mod1<-gamlss(dat1~1,family=BE) # fits a constant for mu and sigma 
#fitted(mod1)[1]
#fitted(mod1,"sigma")[1]
plot(function(y) dBE(y, mu=.1 ,sigma=.5), 0.001, .999)
plot(function(y) pBE(y, mu=.1 ,sigma=.5), 0.001, 0.999)
plot(function(y) qBE(y, mu=.1 ,sigma=.5), 0.001, 0.999)
plot(function(y) qBE(y, mu=.1 ,sigma=.5, lower.tail=FALSE), 0.001, .999)
dat2<-rBEo(100, mu=1, sigma=2)
#mod2<-gamlss(dat2~1,family=BEo) # fits a constant for mu and sigma 
#fitted(mod2)[1]
#fitted(mod2,"sigma")[1]

The beta inflated distribution for fitting a GAMLSS

Description

The function BEINF() defines the beta inflated distribution, a four parameter distribution, for a gamlss.family object to be used in GAMLSS fitting using the function gamlss(). The beta inflated is similar to the beta but allows zeros and ones as values for the response variable. The two extra parameters model the probabilities at zero and one.

The functions BEINF0() and BEINF1() are three parameter beta inflated distributions allowing zeros or ones only at the response respectively. BEINF0() and BEINF1() are re-parameterize versions of the distributions BEZI and BEOI contributed to gamlss by Raydonal Ospina (see Ospina and Ferrari (2010)).

The functions dBEINF, pBEINF, qBEINF and rBEINF define the density, distribution function, quantile function and random generation for the BEINF parametrization of the beta inflated distribution.

The functions dBEINF0, pBEINF0, qBEINF0 and rBEINF0 define the density, distribution function, quantile function and random generation for the BEINF0 parametrization of the beta inflated at zero distribution.

The functions dBEINF1, pBEINF1, qBEINF1 and rBEINF1 define the density, distribution function, quantile function and random generation for the BEINF1 parametrization of the beta inflated at one distribution.

plotBEINF, plotBEINF0 and plotBEINF1 can be used to plot the distributions. meanBEINF, meanBEINF0 and meanBEINF1 calculates the expected value of the response for a fitted model.

Usage

BEINF(mu.link = "logit", sigma.link = "logit", nu.link = "log", 
      tau.link = "log")
BEINF0(mu.link = "logit", sigma.link = "logit", nu.link = "log")
BEINF1(mu.link = "logit", sigma.link = "logit", nu.link = "log")
      
dBEINF(x, mu = 0.5, sigma = 0.1, nu = 0.1, tau = 0.1, 
       log = FALSE)
dBEINF0(x, mu = 0.5, sigma = 0.1, nu = 0.1, log = FALSE)
dBEINF1(x, mu = 0.5, sigma = 0.1, nu = 0.1, log = FALSE)       

pBEINF(q, mu = 0.5, sigma = 0.1, nu = 0.1, tau = 0.1, 
       lower.tail = TRUE, log.p = FALSE)
pBEINF0(q, mu = 0.5, sigma = 0.1, nu = 0.1, 
       lower.tail = TRUE, log.p = FALSE)       
pBEINF1(q, mu = 0.5, sigma = 0.1, nu = 0.1, 
        lower.tail = TRUE, log.p = FALSE)

qBEINF(p, mu = 0.5, sigma = 0.1, nu = 0.1, tau = 0.1, 
       lower.tail = TRUE, log.p = FALSE)
qBEINF0(p, mu = 0.5, sigma = 0.1, nu = 0.1, tau = 0.1, 
        lower.tail = TRUE, log.p = FALSE)
qBEINF1(p, mu = 0.5, sigma = 0.1, nu = 0.1, 
        lower.tail = TRUE, log.p = FALSE)
       
rBEINF(n, mu = 0.5, sigma = 0.1, nu = 0.1, tau = 0.1)
rBEINF0(n, mu = 0.5, sigma = 0.1, nu = 0.1)
rBEINF1(n, mu = 0.5, sigma = 0.1, nu = 0.1)

plotBEINF(mu = 0.5, sigma = 0.5, nu = 0.5, tau = 0.5, 
          from = 0.001, to = 0.999, n = 101, ...)
plotBEINF0(mu = 0.5, sigma = 0.5, nu = 0.5, 
          from = 1e-04, to = 0.9999, n = 101, ...)
plotBEINF1(mu = 0.5, sigma = 0.5, nu = 0.5, 
          from = 1e-04, to = 0.9999, n = 101, ...)

meanBEINF(obj)
meanBEINF0(obj)
meanBEINF1(obj)

Arguments

Details

The beta inflated distribution is defined as

f(y)=p_0 \quad \textit{if} \quad y=0

f(y)=p_1 \quad \textit{if} \quad y=1

f(y|\alpha,\beta)=\frac{1}{B(\alpha, \beta)} y^{\alpha-1}(1-y)^{\beta-1} \quad \textit{otherwise}

for 0 \le y \le 1, \alpha>0, \beta>0, 0<p_0<1, 0<p_1<1 and 0<p_0+p_1<1.

The GAMLSS function BEINF() re-parametrize the parameters as

\mu=\frac{\alpha}{\alpha+\beta}

\sigma=\frac{1}{\alpha+\beta+1}

\nu=\frac{p_0}{p_2}

\tau=\frac{p_1}{p_2}

where p_2=1-p_0-p_1 so 0<\mu<1, 0<\sigma<1, \nu>0 and \tau>0 see pp. 466-467 of Rigby et al. (2019).

The beta inflated at zero distribution is defined as

f(y)=p_0 \quad \textit{if} \quad y=0

f(y|\alpha,\beta)=\frac{1}{B(\alpha, \beta)} y^{\alpha-1}(1-y)^{\beta-1} \quad \textit{otherwise}

for 0 \le y < 1, \alpha>0, \beta>0 and 0<p_0<1.

The GAMLSS function BEINF0() re-parametrize the parameters as

\mu=\frac{\alpha}{\alpha+\beta}

\sigma=\frac{1}{\alpha+\beta+1}

\nu=\frac{p_0}{1-P_0}

so 0<\mu<1, 0<\sigma<1 and \nu>0 see pp. 467-468 of Rigby et al. (2019).

The beta inflated at 1 distribution is defined as

f(y|\alpha,\beta)=\frac{1}{B(\alpha, \beta)} y^{\alpha-1}(1-y)^{\beta-1} \quad \textit{if} \quad 0<y<1

f(y)=p_1 \quad \quad \textit{if} \: \quad y=1

for 0 < y \le 1, \alpha>0, \beta>0 and 0<p_1<1.

The GAMLSS function BEINF1() re-parametrize the parameters as

\mu=\frac{\alpha}{\alpha+\beta}

\sigma=\frac{1}{\alpha+\beta+1}

\nu=\frac{p_1}{1-P_1}

so 0<\mu<1, 0<\sigma<1 and \nu>0 see pp. 468-469 of Rigby et al. (2019).

Value

returns a gamlss.family object which can be used to fit a beta inflated distribution in the gamlss() function. ...

Author(s)

Bob Rigby and Mikis Stasinopoulos

References

Ospina R. and Ferrari S. L. P. (2010) Inflated beta distributions, Statistical Papers, 23, 111-126.

Rigby, R. A. and Stasinopoulos D. M. (2005). Generalized additive models for location, scale and shape,(with discussion), Appl. Statist., 54, part 3, pp 507-554.

Rigby, R. A., Stasinopoulos, D. M., Heller, G. Z., and De Bastiani, F. (2019) Distributions for modeling location, scale, and shape: Using GAMLSS in R, Chapman and Hall/CRC,doi:10.1201/9780429298547. An older version can be found in https://www.gamlss.com/.

Stasinopoulos D. M. Rigby R.A. (2007) Generalized additive models for location scale and shape (GAMLSS) in R. Journal of Statistical Software, Vol. 23, Issue 7, Dec 2007, doi:10.18637/jss.v023.i07.

Stasinopoulos D. M., Rigby R.A., Heller G., Voudouris V., and De Bastiani F., (2017) Flexible Regression and Smoothing: Using GAMLSS in R, Chapman and Hall/CRC. doi:10.1201/b21973

(see also https://www.gamlss.com/).

See Also

gamlss.family, BE, BEo, BEZI, BEOI

Examples

BEINF()# gives information about the default links for the beta inflated distribution
BEINF0()
BEINF1()
# plotting the distributions
op<-par(mfrow=c(2,2)) 
plotBEINF( mu =.5 , sigma=.5, nu = 0.5, tau = 0.5, from = 0, to=1, n = 101)
plotBEINF0( mu =.5 , sigma=.5, nu = 0.5,  from = 0, to=1, n = 101)
plotBEINF1( mu =.5 , sigma=.5, nu = 0.5,  from = 0.001, to=1, n = 101)
curve(dBE(x, mu =.5, sigma=.5),  0.01, 0.999)
par(op)
# plotting the cdf
op<-par(mfrow=c(2,2)) 
plotBEINF( mu =.5 , sigma=.5, nu = 0.5, tau = 0.5, from = 0, to=1, n = 101, main="BEINF")
plotBEINF0( mu =.5 , sigma=.5, nu = 0.5,  from = 0, to=1, n = 101, main="BEINF0")
plotBEINF1( mu =.5 , sigma=.5, nu = 0.5,  from = 0.001, to=1, n = 101, main="BEINF1")
curve(dBE(x, mu =.5, sigma=.5),  0.01, 0.999, main="BE")
par(op)
#---------------------------------------------
op<-par(mfrow=c(2,2)) 
plotBEINF( mu =.5 , sigma=.5, nu = 0.5, tau = 0.5, from = 0, to=1, n = 101, main="BEINF")
plotBEINF0( mu =.5 , sigma=.5, nu = 0.5,  from = 0, to=1, n = 101, main="BEINF0")
plotBEINF1( mu =.5 , sigma=.5, nu = 0.5,  from = 0.001, to=1, n = 101, main="BEINF1")
curve(dBE(x, mu =.5, sigma=.5),  0.01, 0.999, main="BE")
par(op)
#---------------------------------------------
op<-par(mfrow=c(2,2)) 
curve( pBEINF(x, mu=.5 ,sigma=.5, nu = 0.5, tau = 0.5,), 0, 1, ylim=c(0,1), main="BEINF" )
curve(pBEINF0(x, mu=.5 ,sigma=.5, nu = 0.5), 0, 1, ylim=c(0,1), main="BEINF0")
curve(pBEINF1(x, mu=.5 ,sigma=.5, nu = 0.5), 0, 1, ylim=c(0,1), main="BEINF1")
curve(    pBE(x, mu=.5 ,sigma=.5), .001, .99, ylim=c(0,1), main="BE")
par(op)
#---------------------------------------------
op<-par(mfrow=c(2,2)) 
curve(qBEINF(x, mu=.5 ,sigma=.5, nu = 0.5, tau = 0.5), .01, .99, main="BEINF" )
curve(qBEINF0(x, mu=.5 ,sigma=.5, nu = 0.5), .01, .99, main="BEINF0" )
curve(qBEINF1(x, mu=.5 ,sigma=.5, nu = 0.5), .01, .99, main="BEINF1" )
curve(qBE(x, mu=.5 ,sigma=.5), .01, .99 , main="BE")
par(op)

#---------------------------------------------
op<-par(mfrow=c(2,2)) 
hist(rBEINF(200, mu=.5 ,sigma=.5, nu = 0.5, tau = 0.5))
hist(rBEINF0(200, mu=.5 ,sigma=.5, nu = 0.5))
hist(rBEINF1(200, mu=.5 ,sigma=.5, nu = 0.5))
hist(rBE(200, mu=.5 ,sigma=.5))
par(op)
# fit a model to the data 
# library(gamlss)
#m1<-gamlss(dat~1,family=BEINF)
#meanBEINF(m1)[1]

The one-inflated beta distribution for fitting a GAMLSS

Description

The function BEOI() defines the one-inflated beta distribution, a three parameter distribution, for a gamlss.family object to be used in GAMLSS fitting using the function gamlss(). The one-inflated beta is similar to the beta distribution but allows ones as y values. This distribution is an extension of the beta distribution using a parameterization of the beta law that is indexed by mean and precision parameters (Ferrari and Cribari-Neto, 2004). The extra parameter models the probability at one. The functions dBEOI, pBEOI, qBEOI and rBEOI define the density, distribution function, quantile function and random generation for the BEOI parameterization of the one-inflated beta distribution. plotBEOI can be used to plot the distribution. meanBEOI calculates the expected value of the response for a fitted model.

Usage

BEOI(mu.link = "logit", sigma.link = "log", nu.link = "logit")

dBEOI(x, mu = 0.5, sigma = 1, nu = 0.1, log = FALSE)

pBEOI(q, mu = 0.5, sigma = 1, nu = 0.1, lower.tail = TRUE, log.p = FALSE)

qBEOI(p, mu = 0.5, sigma = 1, nu = 0.1, lower.tail = TRUE,
        log.p = FALSE)
        
rBEOI(n, mu = 0.5, sigma = 1, nu = 0.1)

plotBEOI(mu = .5, sigma = 1, nu = 0.1, from = 0.001, to = 1, n = 101, 
    ...)
    
meanBEOI(obj)

Arguments

Details

The one-inflated beta distribution is given as

f(y)=\nu \quad \textit{if} \quad y=1

f(y|\mu,\sigma)=(1-\nu)\frac{\Gamma(\sigma)}{\Gamma(\mu \sigma)\Gamma((1-\mu) \sigma)} y^{\mu \sigma}(1-y)^{((1-\mu)\sigma)-1} \quad \textit{otherwise}

The parameters satisfy 0<\mu<0, \sigma>0 and 0<\nu< 1.

Here E(y)=\nu+(1-\nu)\mu and Var(y)=(1-\nu)\frac{\mu(1-\mu)}{\sigma+1}+\nu(1-\nu)(1-\mu)^2.

Value

returns a gamlss.family object which can be used to fit a one-inflated beta distribution in the gamlss() function.

Note

This work is part of my PhD project at the University of Sao Paulo under the supervion of Professor Silvia Ferrari. My thesis is concerned with regression modelling of rates and proportions with excess of zeros and/or ones

Author(s)

Raydonal Ospina, Department of Statistics, University of Sao Paulo, Brazil.

rospina@ime.usp.br

References

Ferrari, S.L.P., Cribari-Neto, F. (2004). Beta regression for modelling rates and proportions. Journal of Applied Statistics, 31 (1), 799-815.

Ospina R. and Ferrari S. L. P. (2010) Inflated beta distributions, Statistical Papers, 23, 111-126.

Rigby, R. A. and Stasinopoulos D. M. (2005). Generalized additive models for location, scale and shape (with discussion). Applied Statistics, 54 (3), 507-554.

Rigby, R. A., Stasinopoulos, D. M., Heller, G. Z., and De Bastiani, F. (2019) Distributions for modeling location, scale, and shape: Using GAMLSS in R, Chapman and Hall/CRC, doi:10.1201/9780429298547. An older version can be found in https://www.gamlss.com/.

Stasinopoulos D. M. Rigby R.A. (2007) Generalized additive models for location scale and shape (GAMLSS) in R. Journal of Statistical Software, Vol. 23, Issue 7, Dec 2007, doi:10.18637/jss.v023.i07.

Stasinopoulos D. M., Rigby R.A., Heller G., Voudouris V., and De Bastiani F., (2017) Flexible Regression and Smoothing: Using GAMLSS in R, Chapman and Hall/CRC. doi:10.1201/b21973

(see also https://www.gamlss.com/).

See Also

gamlss.family, BEOI

Examples

BEOI()# gives information about the default links for the BEOI distribution
# plotting the distribution
plotBEOI( mu =0.5 , sigma=5, nu = 0.1, from = 0.001, to=1, n = 101)
# plotting the cdf
plot(function(y) pBEOI(y, mu=.5 ,sigma=5, nu=0.1), 0.001, 0.999)
# plotting the inverse cdf
plot(function(y) qBEOI(y, mu=.5 ,sigma=5, nu=0.1), 0.001, 0.999)
# generate random numbers
dat<-rBEOI(100, mu=.5, sigma=5, nu=0.1)
# fit a model to the data. 
# library(gamlss)
#mod1<-gamlss(dat~1,sigma.formula=~1, nu.formula=~1, family=BEOI) 
#fitted(mod1)[1]
#summary(mod1)
#fitted(mod1,"mu")[1]        #fitted mu
#fitted(mod1,"sigma")[1]     #fitted sigma
#fitted(mod1,"nu")[1]        #fitted nu
#meanBEOI(mod1)[1] # expected value of the response

The zero-inflated beta distribution for fitting a GAMLSS

Description

The function BEZI() defines the zero-inflated beta distribution, a three parameter distribution, for a gamlss.family object to be used in GAMLSS fitting using the function gamlss(). The zero-inflated beta is similar to the beta distribution but allows zeros as y values. This distribution is an extension of the beta distribution using a parameterization of the beta law that is indexed by mean and precision parameters (Ferrari and Cribari-Neto, 2004). The extra parameter models the probability at zero. The functions dBEZI, pBEZI, qBEZI and rBEZI define the density, distribution function, quantile function and random generation for the BEZI parameterization of the zero-inflated beta distribution. plotBEZI can be used to plot the distribution. meanBEZI calculates the expected value of the response for a fitted model.

Usage

BEZI(mu.link = "logit", sigma.link = "log", nu.link = "logit")

dBEZI(x, mu = 0.5, sigma = 1, nu = 0.1, log = FALSE)

pBEZI(q, mu = 0.5, sigma = 1, nu = 0.1, lower.tail = TRUE, log.p = FALSE)

qBEZI(p, mu = 0.5, sigma = 1, nu = 0.1, lower.tail = TRUE,
        log.p = FALSE)
        
rBEZI(n, mu = 0.5, sigma = 1, nu = 0.1)

plotBEZI(mu = .5, sigma = 1, nu = 0.1, from = 0, to = 0.999, n = 101, 
    ...)
    
meanBEZI(obj)

Arguments

Details

The zero-inflated beta distribution is given as

f(y)=\nu

if (y=0)

f(y|\mu,\sigma)=(1-\nu)\frac{\Gamma(\sigma)}{\Gamma(\mu\sigma)\Gamma((1-\mu)\sigma)} y^{\mu\sigma}(1-y)^{((1-\mu)\sigma)-1}

if y=(0,1). The parameters satisfy 0<\mu<0, \sigma>0 and 0<\nu< 1.

Here E(y)=(1-\nu)\mu and Var(y)=(1-\nu)\frac{\mu(1-\mu)}{\sigma+1}+\nu(1-\nu)\mu^2.

Value

returns a gamlss.family object which can be used to fit a zero-inflated beta distribution in the gamlss() function.

Note

This work is part of my PhD project at the University of Sao Paulo under the supervion of Professor Silvia Ferrari. My thesis is concerned with regression modelling of rates and proportions with excess of zeros and/or ones

Author(s)

Raydonal Ospina, Department of Statistics, University of Sao Paulo, Brazil.

rospina@ime.usp.br

References

Ferrari, S.L.P., Cribari-Neto, F. (2004). Beta regression for modelling rates and proportions. Journal of Applied Statistics, 31 (1), 799-815.

Ospina R. and Ferrari S. L. P. (2010) Inflated beta distributions, Statistical Papers, 23, 111-126.

Rigby, R. A. and Stasinopoulos D. M. (2005). Generalized additive models for location, scale and shape (with discussion). Applied Statistics, 54 (3), 507-554.

Rigby, R. A., Stasinopoulos, D. M., Heller, G. Z., and De Bastiani, F. (2019) Distributions for modeling location, scale, and shape: Using GAMLSS in R, Chapman and Hall/CRC, doi:10.1201/9780429298547. An older version can be found in https://www.gamlss.com/.

Stasinopoulos D. M. Rigby R.A. (2007) Generalized additive models for location scale and shape (GAMLSS) in R. Journal of Statistical Software, Vol. 23, Issue 7, Dec 2007, doi:10.18637/jss.v023.i07.

Stasinopoulos D. M., Rigby R.A., Heller G., Voudouris V., and De Bastiani F., (2017) Flexible Regression and Smoothing: Using GAMLSS in R, Chapman and Hall/CRC. doi:10.1201/b21973

(see also https://www.gamlss.com/).

See Also

gamlss.family, BEZI

Examples

 BEZI()# gives information about the default links for the BEZI distribution
# plotting the distribution
plotBEZI( mu =0.5 , sigma=5, nu = 0.1, from = 0, to=0.99, n = 101)
# plotting the cdf
plot(function(y) pBEZI(y, mu=.5 ,sigma=5, nu=0.1), 0, 0.999)
# plotting the inverse cdf
plot(function(y) qBEZI(y, mu=.5 ,sigma=5, nu=0.1), 0, 0.999)
# generate random numbers
dat<-rBEZI(100, mu=.5, sigma=5, nu=0.1)
# fit a model to the data. Tits a constant for mu, sigma and nu
# library(gamlss)
#mod1<-gamlss(dat~1,sigma.formula=~1, nu.formula=~1, family=BEZI) 
#fitted(mod1)[1]
#summary(mod1)
#fitted(mod1,"mu")[1]         #fitted mu   
#fitted(mod1,"sigma")[1]      #fitted sigma 
#fitted(mod1,"nu")[1]         #fitted nu  
#meanBEZI(mod1)[1] # expected value of the response

Binomial distribution for fitting a GAMLSS

Description

The BI() function defines the binomial distribution, a one parameter family distribution, for a gamlss.family object to be used in GAMLSS fitting using the function gamlss(). The functions dBI, pBI, qBI and rBI define the density, distribution function, quantile function and random generation for the binomial, BI(), distribution.

Usage

BI(mu.link = "logit")
dBI(x, bd = 1, mu = 0.5, log = FALSE)
pBI(q, bd = 1, mu = 0.5, lower.tail = TRUE, log.p = FALSE)
qBI(p, bd = 1, mu = 0.5, lower.tail = TRUE, log.p = FALSE)
rBI(n, bd = 1, mu = 0.5)

Arguments

Details

Definition file for binomial distribution.

f(y|\mu)=\frac{\Gamma(n+1)}{\Gamma(y+1) \Gamma{(n-y+1)}} \mu^y (1-\mu)^{(n-y)}

for y=0,1,2,...,n and 0<\mu< 1 see pp. 521-522 of Rigby et al. (2019).

Value

returns a gamlss.family object which can be used to fit a binomial distribution in the gamlss() function.

Note

The response variable should be a matrix containing two columns, the first with the count of successes and the second with the count of failures. The parameter mu represents a probability parameter with limits 0 < \mu < 1. n\mu is the mean of the distribution where n is the binomial denominator.

Author(s)

Mikis Stasinopoulos, Bob Rigby and Calliope Akantziliotou

References

Rigby, R. A. and Stasinopoulos D. M. (2005). Generalized additive models for location, scale and shape,(with discussion), Appl. Statist., 54, part 3, pp 507-554.

Rigby, R. A., Stasinopoulos, D. M., Heller, G. Z., and De Bastiani, F. (2019) Distributions for modeling location, scale, and shape: Using GAMLSS in R, Chapman and Hall/CRC, doi:10.1201/9780429298547. An older version can be found in https://www.gamlss.com/.

Stasinopoulos D. M. Rigby R.A. (2007) Generalized additive models for location scale and shape (GAMLSS) in R. Journal of Statistical Software, Vol. 23, Issue 7, Dec 2007, doi:10.18637/jss.v023.i07.

Stasinopoulos D. M., Rigby R.A., Heller G., Voudouris V., and De Bastiani F., (2017) Flexible Regression and Smoothing: Using GAMLSS in R, Chapman and Hall/CRC. doi:10.1201/b21973

(see also https://www.gamlss.com/).

See Also

gamlss.family, ZABI, ZIBI

Examples

 BI()# gives information about the default links for the Binomial distribution 
# data(aep)   
# library(gamlss)
# h<-gamlss(y~ward+loglos+year, family=BI, data=aep)  
# plot of the binomial distribution
curve(dBI(x, mu = .5, bd=10), from=0, to=10, n=10+1, type="h")
tN <- table(Ni <- rBI(1000, mu=.2, bd=10))
r <- barplot(tN, col='lightblue')

Beta Negative Binomial distribution for fitting a GAMLSS

Description

The BNB() function defines the beta negative binomial distribution, a three parameter distribution, for a gamlss.family object to be used in GAMLSS fitting using the function gamlss().

The functions dBNB, pBNB, qBNB and rBNB define the density, distribution function, quantile function and random generation for the beta negative binomial distribution, BNB().

The functions ZABNB() and ZIBNB() are the zero adjusted (hurdle) and zero inflated versions of the beta negative binomial distribution, respectively. That is four parameter distributions.

The functions dZABNB, dZIBNB, pZABNB,pZIBNB, qZABNB qZIBNB rZABNB and rZIBNB define the probability, cumulative, quantile and random generation functions for the zero adjusted and zero inflated beta negative binomial distributions, ZABNB(), ZIBNB(), respectively.

Usage

BNB(mu.link = "log", sigma.link = "log", nu.link = "log")
dBNB(x, mu = 1, sigma = 1, nu = 1, log = FALSE)
pBNB(q, mu = 1, sigma = 1, nu = 1, lower.tail = TRUE, log.p = FALSE)
qBNB(p, mu = 1, sigma = 1, nu = 1, lower.tail = TRUE, log.p = FALSE, 
     max.value = 10000)
rBNB(n, mu = 1, sigma = 1, nu = 1, max.value = 10000)

ZABNB(mu.link = "log", sigma.link = "log", nu.link = "log",
      tau.link = "logit")
dZABNB(x, mu = 1, sigma = 1, nu = 1, tau = 0.1, log = FALSE)
pZABNB(q, mu = 1, sigma = 1, nu = 1, tau = 0.1, lower.tail = TRUE, 
       log.p = FALSE)
qZABNB(p, mu = 1, sigma = 1, nu = 1, tau = 0.1, lower.tail = TRUE, 
       log.p = FALSE, max.value = 10000)
rZABNB(n, mu = 1, sigma = 1, nu = 1, tau = 0.1, max.value = 10000)

ZIBNB(mu.link = "log", sigma.link = "log", nu.link = "log", 
      tau.link = "logit")
dZIBNB(x, mu = 1, sigma = 1, nu = 1, tau = 0.1, log = FALSE)
pZIBNB(q, mu = 1, sigma = 1, nu = 1, tau = 0.1, lower.tail = TRUE, 
       log.p = FALSE)
qZIBNB(p, mu = 1, sigma = 1, nu = 1, tau = 0.1, lower.tail = TRUE, 
       log.p = FALSE, max.value = 10000)
rZIBNB(n, mu = 1, sigma = 1, nu = 1, tau = 0.1, max.value = 10000)       

Arguments

Details

The probability function of the BNB is

P(Y=y|\mu,\sigma, \nu) = \frac{\Gamma(y+\nu^{-1}) B(y+\mu \sigma^{-1} \nu,\sigma^{-1}+\nu^{-1}+1)} {\Gamma(y+1) \Gamma(\nu^{-1}) B(\mu \sigma^{-1} \nu,\sigma^{-1}+1)}

for y=0,1,2,3,..., \mu>0, \sigma>0 and \nu>0, see pp 502-503 of Rigby et al. (2019).

The distribution has mean \mu.

The definition of the zero adjusted beta negative binomial distribution, ZABNB and the the zero inflated beta negative binomial distribution, ZIBNB, are given in p. 517 and pp. 519 of of Rigby et al. (2019), respectively.

Value

returns a gamlss.family object which can be used to fit a Poisson distribution in the gamlss() function.

Author(s)

Bob Rigby and Mikis Stasinopoulos

References

Rigby, R. A. and Stasinopoulos D. M. (2005). Generalized additive models for location, scale and shape,(with discussion), Appl. Statist., 54, part 3, pp 507-554.

Rigby, R. A., Stasinopoulos, D. M., Heller, G. Z., and De Bastiani, F. (2019) Distributions for modeling location, scale, and shape: Using GAMLSS in R, Chapman and Hall/CRC, doi:10.1201/9780429298547. An older version can be found in https://www.gamlss.com/.

Stasinopoulos D. M., Rigby R.A. and Akantziliotou C. (2006) Instructions on how to use the GAMLSS package in R. Accompanying documentation in the current GAMLSS help files, (see also https://www.gamlss.com/).

Stasinopoulos D. M. Rigby R.A. (2007) Generalized additive models for location scale and shape (GAMLSS) in R. Journal of Statistical Software, Vol. 23, Issue 7, Dec 2007, doi:10.18637/jss.v023.i07.

Stasinopoulos D. M., Rigby R.A., Heller G., Voudouris V., and De Bastiani F., (2017) Flexible Regression and Smoothing: Using GAMLSS in R, Chapman and Hall/CRC. doi:10.1201/b21973

(see also https://www.gamlss.com/).

See Also

NBI, NBII

Examples

BNB()   # gives information about the default links for the beta negative binomial
# plotting the distribution
plot(function(y) dBNB(y, mu = 10, sigma = 0.5, nu=2), from=0, to=40, n=40+1, type="h")
# creating random variables and plot them 
tN <- table(Ni <- rBNB(1000, mu=5, sigma=0.5, nu=2))
r <- barplot(tN, col='lightblue')

ZABNB()
ZIBNB()
# plotting the distribution
plot(function(y) dZABNB(y, mu = 10, sigma = 0.5, nu=2, tau=.1),  
     from=0, to=40, n=40+1, type="h")
plot(function(y) dZIBNB(y, mu = 10, sigma = 0.5, nu=2, tau=.1),  
     from=0, to=40, n=40+1, type="h")
## Not run: 
library(gamlss)
data(species)
species <- transform(species, x=log(lake))
m6 <- gamlss(fish~ pb(x), sigma.fo=~1, data=species, family=BNB)

## End(Not run)

The Double binomial distribution

Description

The function DBI() defines the double binomial distribution, a two parameters distribution, for a gamlss.family object to be used in GAMLSS fitting using the function gamlss(). The functions dDBI, pDBI, qDBI and rDBI define the density, distribution function, quantile function and random generation for the double binomial, DBI(), distribution. The function GetBI_C calculates numericaly the constant of proportionality needed for the pdf to sum up to 1.

Usage

DBI(mu.link = "logit", sigma.link = "log")
dDBI(x, mu = 0.5, sigma = 1, bd = 2, log = FALSE)
pDBI(q, mu = 0.5, sigma = 1, bd = 2, lower.tail = TRUE, 
       log.p = FALSE)
qDBI(p, mu = 0.5, sigma = 1, bd = 2, lower.tail = TRUE, 
       log.p = FALSE)
rDBI(n, mu = 0.5, sigma = 1, bd = 2)       
GetBI_C(mu, sigma, bd)       

Arguments

Details

The definition for the Double Poisson distribution first introduced by Efron (1986) is:

p_Y(y|n, \mu,\sigma)= \frac{1}{C(n,\mu,\sigma)} \frac{\Gamma(n+1)}{\Gamma(y+1)\Gamma(n-y+1)} \frac{y^y \left(n-y \right)^{n-y}}{n^n} \frac{n^{n/\sigma} \mu^{y/\sigma} \left( 1-\mu\right)^{(n-y)/\sigma}} {y^{y/\sigma} \left( n-y\right)^{(n-y)/\sigma}}

for y=0,1,2,\ldots,\infty, \mu>0 and \sigma>0 where C is the constant of proportinality which is calculated numerically using the function GetBI_C(), see pp. 524-525 of Rigby et al. (2019).

Value

The function DBI returns a gamlss.family object which can be used to fit a double binomial distribution in the gamlss() function.

Author(s)

Mikis Stasinopoulos, Bob Rigby, Marco Enea and Fernanda de Bastiani

References

Efron, B., 1986. Double exponential families and their use in generalized linear Regression. Journal of the American Statistical Association 81 (395), 709-721.

Rigby, R. A. and Stasinopoulos D. M. (2005). Generalized additive models for location, scale and shape,(with discussion), Appl. Statist., 54, part 3, pp 507-554.

Rigby, R. A., Stasinopoulos, D. M., Heller, G. Z., and De Bastiani, F. (2019) Distributions for modeling location, scale, and shape: Using GAMLSS in R, Chapman and Hall/CRC, doi:10.1201/9780429298547. An older version can be found in https://www.gamlss.com/.

Stasinopoulos D. M. Rigby R.A. (2007) Generalized additive models for location scale and shape (GAMLSS) in R. Journal of Statistical Software, Vol. 23, Issue 7, Dec 2007, doi:10.18637/jss.v023.i07.

Stasinopoulos D. M., Rigby R.A., Heller G., Voudouris V., and De Bastiani F., (2017) Flexible Regression and Smoothing: Using GAMLSS in R, Chapman and Hall/CRC. doi:10.1201/b21973

(see also https://www.gamlss.com/).

See Also

BI,BB

Examples

DBI()
x <- 0:20
# underdispersed DBI
plot(x, dDBI(x, mu=.5, sigma=.2, bd=20), type="h", col="green", lwd=2)
# binomial
lines(x+0.1, dDBI(x, mu=.5, sigma=1, bd=20), type="h", col="black", lwd=2)
# overdispersed DBI
lines(x+.2, dDBI(x, mu=.5, sigma=2, bd=20), type="h", col="red",lwd=2)

The Discrete Burr type XII distribution for fitting a GAMLSS model

Description

The DBURR12() function defines the discrete Burr type XII distribution, a three parameter discrete distribution, for a gamlss.family object to be used in GAMLSS fitting using the function gamlss(). The functions dDBURR12(), pDBURR12(), qDBURR12() and rDBURR12() define the density, distribution function, quantile function and random generation for the discrete Burr type XII DBURR12(), distribution.

Usage

DBURR12(mu.link = "log", sigma.link = "log", nu.link = "log")
dDBURR12(x, mu = 5, sigma = 2, nu = 2, log = FALSE)
pDBURR12(q, mu = 5, sigma = 2, nu = 2, lower.tail = TRUE, 
        log.p = FALSE)
qDBURR12(p, mu = 5, sigma = 2, nu = 2, lower.tail = TRUE, 
        log.p = FALSE)
rDBURR12(n, mu = 5, sigma = 2, nu = 2)

Arguments

Details

The probability function of the discrete Burr XII distribution is given by

f(y|\mu,\sigma,\nu)= (1+(y/\mu)^\sigma)^\nu - (1+((y+1)/\mu)^\sigma)^\nu

for y=0,1,2,...,\infty, \mu>0 , \sigma>0 and \mu>0 see pp 504-505 of Rigby et al. (2019).

Note that the above parametrization is different from Para and Jan (2016).

Value

The function DBURR12() Returns a gamlss.family object which can be used to fit a discrete Burr XII distribution in the gamlss() function.

Note

The parameters of the distributioins are highly correlated so the argument of gamlss method=mixed(10,100) may have to be used.

The distribution can be under/over dispersed and also with long tails.

Author(s)

Rigby, R. A., Stasinopoulos D. M., Fernanda De Bastiani.

References

Para, B. A. and Jan, T. R. (2016). On discrete three parameter Burr type XII and discrete Lomax distributions and their applications to model count data from medical science. Biometrics and Biostatistics International Journal, 54, part 3, pp 507-554.

Rigby, R. A. and Stasinopoulos D. M. (2005). Generalized additive models for location, scale and shape,(with discussion), Appl. Statist., 4, pp 1-15.

Rigby, R. A., Stasinopoulos, D. M., Heller, G. Z., and De Bastiani, F. (2019) Distributions for modeling location, scale, and shape: Using GAMLSS in R, Chapman and Hall/CRC, doi:10.1201/9780429298547. An older version can be found in https://www.gamlss.com/.

Stasinopoulos D. M. Rigby R.A. (2007) Generalized additive models for location scale and shape (GAMLSS) in R. Journal of Statistical Software, Vol. 23, Issue 7, Dec 2007, doi:10.18637/jss.v023.i07.

Stasinopoulos D. M., Rigby R.A., Heller G., Voudouris V., and De Bastiani F., (2017) Flexible Regression and Smoothing: Using GAMLSS in R, Chapman and Hall/CRC. doi:10.1201/b21973

(see also https://www.gamlss.com/).

See Also

gamlss.family, DPO

Examples

DBURR12()# 
#plot the pdf using plot 
plot(function(y) dDBURR12(y, mu=10, sigma=1, nu=1), from=0, to=100, n=100+1, type="h") # pdf
# plot the cdf
plot(seq(from=0,to=100),pDBURR12(seq(from=0,to=100), mu=10, sigma=1, nu=1), type="h")   # cdf
# generate random sample
tN <- table(Ni <- rDBURR12(100, mu=5, sigma=1, nu=1))
r <- barplot(tN, col='lightblue')

The Delaporte distribution for fitting a GAMLSS model

Description

The DEL() function defines the Delaporte distribution, a three parameter discrete distribution, for a gamlss.family object to be used in GAMLSS fitting using the function gamlss(). The functions dDEL, pDEL, qDEL and rDEL define the density, distribution function, quantile function and random generation for the Delaporte DEL(), distribution.

Usage

DEL(mu.link = "log", sigma.link = "log", nu.link = "logit")
dDEL(x, mu=1, sigma=1, nu=0.5, log=FALSE)
pDEL(q, mu=1, sigma=1, nu=0.5, lower.tail = TRUE, 
        log.p = FALSE)
qDEL(p, mu=1, sigma=1, nu=0.5,  lower.tail = TRUE, 
     log.p = FALSE,  max.value = 10000)        
rDEL(n, mu=1, sigma=1, nu=0.5, max.value = 10000)

Arguments

Details

The probability function of the Delaporte distribution is given by

f(y|\mu,\sigma,\nu)= \frac{e^{-\mu \nu}}{\Gamma(1/\sigma)}\left[ 1+\mu \sigma (1-\nu)\right]^{-1/\sigma} S

where

S= \sum_{j=0}^{y} \left( { y \choose j } \right) \frac{\mu^y \nu^{y-j}}{y!}\left[\mu + \frac{1}{\sigma(1-\nu)}\right]^{-j} \Gamma\left(\frac{1}{\sigma}+j\right)

for y=0,1,2,...,\infty where \mu>0 , \sigma>0 and 0 < \nu<1. This distribution is a parametrization of the distribution given by Wimmer and Altmann (1999) p 515-516 where \alpha=\mu \nu, k=1/\sigma and \rho=[1+\mu\sigma(1-\nu)]^{-1}. For more details see pp 506-507 of Rigby et al. (2019).

Value

Returns a gamlss.family object which can be used to fit a Delaporte distribution in the gamlss() function.

Note

The mean of Y is given by E(Y)=\mu and the variance by V(Y)=\mu+\mu^2 \sigma \left( 1-\nu\right)^2.

Author(s)

Rigby, R. A., Stasinopoulos D. M. and Marco Enea

References

Rigby, R. A. and Stasinopoulos D. M. (2005). Generalized additive models for location, scale and shape,(with discussion), Appl. Statist., 54, part 3, pp 507-554.

Rigby, R. A., Stasinopoulos, D. M., Heller, G. Z., and De Bastiani, F. (2019) Distributions for modeling location, scale, and shape: Using GAMLSS in R, Chapman and Hall/CRC,doi:10.1201/9780429298547. An older version can be found in https://www.gamlss.com/.

Stasinopoulos D. M. Rigby R.A. (2007) Generalized additive models for location scale and shape (GAMLSS) in R. Journal of Statistical Software, Vol. 23, Issue 7, Dec 2007, doi:10.18637/jss.v023.i07.

Stasinopoulos D. M., Rigby R.A., Heller G., Voudouris V., and De Bastiani F., (2017) Flexible Regression and Smoothing: Using GAMLSS in R, Chapman and Hall/CRC. doi:10.1201/b21973

Wimmer, G. and Altmann, G (1999). Thesaurus of univariate discrete probability distributions . Stamn Verlag, Essen, Germany

(see also https://www.gamlss.com/).

See Also

gamlss.family, SI , SICHEL

Examples

 DEL()# gives information about the default links for the  Delaporte distribution 
#plot the pdf using plot 
plot(function(y) dDEL(y, mu=10, sigma=1, nu=.5), from=0, to=100, n=100+1, type="h") # pdf
# plot the cdf
plot(seq(from=0,to=100),pDEL(seq(from=0,to=100), mu=10, sigma=1, nu=0.5), type="h")   # cdf
# generate random sample
tN <- table(Ni <- rDEL(100, mu=10, sigma=1, nu=0.5))
r <- barplot(tN, col='lightblue')
# fit a model to the data 
# libary(gamlss)
# gamlss(Ni~1,family=DEL, control=gamlss.control(n.cyc=50))

The Double Poisson distribution

Description

The function DPO() defines the double Poisson distribution, a two parameters distribution, for a gamlss.family object to be used in GAMLSS fitting using the function gamlss(). The functions dDPO, pDPO, qDPO and rPO define the density, distribution function, quantile function and random generation for the double Poisson, DPO(), distribution. The function get_C() calculates numericaly the constant of proportionality needed for the pdf to sum up to 1.

Usage

DPO(mu.link = "log", sigma.link = "log")
dDPO(x, mu = 1, sigma = 1, log = FALSE)
pDPO(q, mu = 1, sigma = 1, lower.tail = TRUE, log.p = FALSE)
qDPO(p, mu = 1, sigma = 1, lower.tail = TRUE, log.p = FALSE, 
     max.value = 10000)
rDPO(n, mu = 1, sigma = 1, max.value = 10000)
get_C(x, mu, sigma)

Arguments

Details

The definition for the Double Poisson distribution first introduced by Efron (1986) is:

f(y|\mu,\sigma)=\left( \frac{1}{\sigma} \right)^{1/2} e^{- \mu / \sigma} \left( \frac{e^{-y} y^{y}}{y!}\right) \left(\frac{e \mu}{y} \right)^{y/\sigma} C

for y=0,1,2,\ldots,\infty, \mu>0 and \sigma>0 where C is the constant of proportinality which is calculated numerically using the function get_C.

Value

The function DPO returns a gamlss.family object which can be used to fit a double Poisson distribution in the gamlss() function.

Note

The distributons calculates the constant of proportionality numerically therefore it can be slow for large data

Author(s)

Mikis Stasinopoulos, Bob Rigby and Marco Enea

References

Efron, B., 1986. Double exponential families and their use in generalized linear Regression. Journal of the American Statistical Association 81 (395), 709-721.

Rigby, R. A. and Stasinopoulos D. M. (2005). Generalized additive models for location, scale and shape,(with discussion), Appl. Statist., 54, part 3, pp 507-554.

Rigby, R. A., Stasinopoulos, D. M., Heller, G. Z., and De Bastiani, F. (2019) Distributions for modeling location, scale, and shape: Using GAMLSS in R, Chapman and Hall/CRC, doi:10.1201/9780429298547. An older version can be found in https://www.gamlss.com/.

Stasinopoulos D. M. Rigby R.A. (2007) Generalized additive models for location scale and shape (GAMLSS) in R. Journal of Statistical Software, Vol. 23, Issue 7, Dec 2007, doi:10.18637/jss.v023.i07.

Stasinopoulos D. M., Rigby R.A., Heller G., Voudouris V., and De Bastiani F., (2017) Flexible Regression and Smoothing: Using GAMLSS in R, Chapman and Hall/CRC. doi:10.1201/b21973

(see also https://www.gamlss.com/).

See Also

PO

Examples

DPO()
# overdisperse DPO
x <- 0:20
plot(x, dDPO(x, mu=5, sigma=3), type="h", col="red")
# underdisperse DPO
plot(x, dDPO(x, mu=5, sigma=.3), type="h", col="red")
# generate random sample
 Y <- rDPO(100,5,.5)
plot(table(Y))
points(0:20, 100*dDPO(0:20, mu=5, sigma=.5)+0.2,  col="red")
# fit a model to the data 
# library(gamlss)
# gamlss(Y~1,family=DPO)

The exponential generalized Beta type 2 distribution for fitting a GAMLSS

Description

This function defines the generalized t distribution, a four parameter distribution. The response variable is in the range from minus infinity to plus infinity. The functions dEGB2, pEGB2, qEGB2 and rEGB2 define the density, distribution function, quantile function and random generation for the generalized beta type 2 distribution.

Usage

EGB2(mu.link = "identity", sigma.link = "log", nu.link = "log", 
      tau.link = "log")
dEGB2(x, mu = 0, sigma = 1, nu = 1, tau = 0.5, log = FALSE)
pEGB2(q, mu = 0, sigma = 1, nu = 1, tau = 0.5, lower.tail = TRUE, 
      log.p = FALSE)
qEGB2(p, mu = 0, sigma = 1, nu = 1, tau = 0.5, lower.tail = TRUE, 
      log.p = FALSE)
rEGB2(n, mu = 0, sigma = 1, nu = 1, tau = 0.5)

Arguments

Details

The probability density function of the Generalized Beta type 2, (GB2), is defined as:

f(y|\mu,\sigma\,\nu,\tau) = e^{\nu z }\left[|\sigma| B(\nu,\tau) \left(1+ e^z\right)^{\nu+\tau} \right]^{-1}

for -\infty<y<\infty, where z=(y-\mu)/\sigma and -\infty<\mu<\infty, -\infty<\sigma<\infty, \nu>0 and \tau>0, McDonald and Xu (1995), see also pp. 385-386 of Rigby et al. (2019).

Value

EGB2() returns a gamlss.family object which can be used to fit the EGB2 distribution in the gamlss() function. dEGB2() gives the density, pEGB2() gives the distribution function, qEGB2() gives the quantile function, and rEGB2() generates random deviates.

Author(s)

Bob Rigby and Mikis Stasinopoulos

References

Rigby, R. A. and Stasinopoulos D. M. (2005). Generalized additive models for location, scale and shape,(with discussion), Appl. Statist., 54, part 3, pp 507-554.

Rigby, R. A., Stasinopoulos, D. M., Heller, G. Z., and De Bastiani, F. (2019) Distributions for modeling location, scale, and shape: Using GAMLSS in R, Chapman and Hall/CRC,doi:10.1201/9780429298547. An older version can be found in https://www.gamlss.com/.

Stasinopoulos D. M. Rigby R.A. (2007) Generalized additive models for location scale and shape (GAMLSS) in R. Journal of Statistical Software, Vol. 23, Issue 7, Dec 2007, doi:10.18637/jss.v023.i07.

Stasinopoulos D. M., Rigby R.A., Heller G., Voudouris V., and De Bastiani F., (2017) Flexible Regression and Smoothing: Using GAMLSS in R, Chapman and Hall/CRC. doi:10.1201/b21973

(see also https://www.gamlss.com/).

See Also

gamlss.family, JSU, BCT

Examples

EGB2()   # 
y<- rEGB2(200, mu=5, sigma=2, nu=1, tau=4)
library(MASS)
truehist(y)
fx<-dEGB2(seq(min(y), 20, length=200), mu=5 ,sigma=2, nu=1, tau=4)
lines(seq(min(y),20,length=200),fx)
# something funny here
# library(gamlss)
# histDist(y, family=EGB2, n.cyc=60)
integrate(function(x) x*dEGB2(x=x, mu=5, sigma=2, nu=1, tau=4), -Inf, Inf)
curve(dEGB2(x, mu=5 ,sigma=2, nu=1, tau=4), -10, 10, main = "The EGB2  density 
             mu=5, sigma=2, nu=1, tau=4")

Exponential distribution for fitting a GAMLSS

Description

The function EXP defines the exponential distribution, a one parameter distribution for a gamlss.family object to be used in GAMLSS fitting using the function gamlss(). The mu parameter represents the mean of the distribution. The functions dEXP, pEXP, qEXP and rEXP define the density, distribution function, quantile function and random generation for the specific parameterization of the exponential distribution defined by function EXP.

Usage

EXP(mu.link ="log")
dEXP(x, mu = 1, log = FALSE)
pEXP(q, mu = 1, lower.tail = TRUE, log.p = FALSE)
qEXP(p, mu = 1, lower.tail = TRUE, log.p = FALSE)
rEXP(n, mu = 1)

Arguments

Details

The specific parameterization of the exponential distribution used in EXP is

f(y|\mu)=\frac{1}{\mu} \exp\left\{-\frac{y}{\mu}\right\}

for y>0, \mu>0 see pp. 422-23 of Rigby et al. (2019).

Value

EXP() returns a gamlss.family object which can be used to fit an exponential distribution in the gamlss() function. dEXP() gives the density, pEXP() gives the distribution function, qEXP() gives the quantile function, and rEXP() generates random deviates.

Author(s)

Mikis Stasinopoulos, Bob Rigby and Nicoleta Motpan

References

Rigby, R. A. and Stasinopoulos D. M. (2005). Generalized additive models for location, scale and shape,(with discussion), Appl. Statist., 54, part 3, pp 507-554.

Rigby, R. A., Stasinopoulos, D. M., Heller, G. Z., and De Bastiani, F. (2019) Distributions for modeling location, scale, and shape: Using GAMLSS in R, Chapman and Hall/CRC,doi:10.1201/9780429298547. An older version can be found in https://www.gamlss.com/.

Stasinopoulos D. M. Rigby R.A. (2007) Generalized additive models for location scale and shape (GAMLSS) in R. Journal of Statistical Software, Vol. 23, Issue 7, Dec 2007, doi:10.18637/jss.v023.i07.

Stasinopoulos D. M., Rigby R.A., Heller G., Voudouris V., and De Bastiani F., (2017) Flexible Regression and Smoothing: Using GAMLSS in R, Chapman and Hall/CRC. doi:10.1201/b21973

(see also https://www.gamlss.com/).

See Also

gamlss.family

Examples

y<-rEXP(1000,mu=1) # generates 1000 random observations 
hist(y)
# library(gamlss)
# histDist(y, family=EXP) 

Gamma distribution for fitting a GAMLSS

Description

The function GA defines the gamma distribution, a two parameter distribution, for a gamlss.family object to be used in GAMLSS fitting using the function gamlss(). The parameterization used has the mean of the distribution equal to \mu and the variance equal to \sigma^2 \mu^2. The functions dGA, pGA, qGA and rGA define the density, distribution function, quantile function and random generation for the specific parameterization of the gamma distribution defined by function GA.

Usage

GA(mu.link = "log", sigma.link ="log")
dGA(x, mu = 1, sigma = 1, log = FALSE) 
pGA(q, mu = 1, sigma = 1, lower.tail = TRUE, log.p = FALSE)
qGA(p, mu = 1, sigma = 1, lower.tail = TRUE, log.p = FALSE)
rGA(n, mu = 1, sigma = 1)

Arguments

Details

The specific parameterization of the gamma distribution used in GA is

f(y|\mu,\sigma)=\frac{y^{(1/\sigma^2-1)}\exp[-y/(\sigma^2 \mu)]}{(\sigma^2 \mu)^{(1/\sigma^2)} \Gamma(1/\sigma^2)}

for y>0, \mu>0 and \sigma>0, see pp. 423-424 of Rigby et al. (2019).

Value

GA() returns a gamlss.family object which can be used to fit a gamma distribution in the gamlss() function. dGA() gives the density, pGA() gives the distribution function, qGA() gives the quantile function, and rGA() generates random deviates. The latest functions are based on the equivalent R functions for gamma distribution.

Note

\mu is the mean of the distribution in GA. In the function GA, \sigma is the square root of the usual dispersion parameter for a GLM gamma model. Hence \sigma \mu is the standard deviation of the distribution defined in GA.

Author(s)

Mikis Stasinopoulos, Bob Rigby and Calliope Akantziliotou

References

Rigby, R. A. and Stasinopoulos D. M. (2005). Generalized additive models for location, scale and shape,(with discussion), Appl. Statist., 54, part 3, pp 507-554.

Rigby, R. A., Stasinopoulos, D. M., Heller, G. Z., and De Bastiani, F. (2019) Distributions for modeling location, scale, and shape: Using GAMLSS in R, Chapman and Hall/CRC,doi:10.1201/9780429298547. An older version can be found in https://www.gamlss.com/.

Stasinopoulos D. M. Rigby R.A. (2007) Generalized additive models for location scale and shape (GAMLSS) in R. Journal of Statistical Software, Vol. 23, Issue 7, Dec 2007, doi:10.18637/jss.v023.i07.

Stasinopoulos D. M., Rigby R.A., Heller G., Voudouris V., and De Bastiani F., (2017) Flexible Regression and Smoothing: Using GAMLSS in R, Chapman and Hall/CRC. doi:10.1201/b21973

(see also https://www.gamlss.com/).

See Also

gamlss.family

Examples

GA()# gives information about the default links for the gamma distribution      
# dat<-rgamma(100, shape=1, scale=10) # generates 100 random observations 
# fit a gamlss model
# gamlss(dat~1,family=GA) 
# fits a constant for each parameter mu and sigma of the gamma distribution
newdata<-rGA(1000,mu=1,sigma=1) # generates 1000 random observations
hist(newdata) 
rm(dat,newdata)

The Gamma distribution family

Description

The function GAF() defines a gamma distribution family, which has three parameters. This is not the generalised gamma distribution which is called GG. The third parameter here is to model the mean and variance relationship. The distribution can be fitted using the function gamlss(). The mean of GAF is equal to mu. The variance is equal to sigma^2*mu^nu so the standard deviation is sigma*mu^(nu/2). The function is design for cases where the variance is proportional to a power of the mean. This is an instance of the Taylor's power low, see Enki et al. (2017). The functions dGAF, pGAF, qGAF and rGAF define the density, distribution function, quantile function and random generation for the GAF parametrization of the gamma family.

Usage

GAF(mu.link = "log", sigma.link = "log", nu.link = "identity")
dGAF(x, mu = 1, sigma = 1, nu = 2, log = FALSE)
pGAF(q, mu = 1, sigma = 1, nu = 2, lower.tail = TRUE, 
    log.p = FALSE)
qGAF(p, mu = 1, sigma = 1, nu = 2, lower.tail = TRUE, 
    log.p = FALSE)
rGAF(n, mu = 1, sigma = 1, nu = 2)

Arguments

Details

The parametrization of the gamma family given in the function GAF() is:

f(y|\mu,\sigma_1)=\frac{y^{(1/\sigma_1^2-1)}\exp[-y/(\sigma_1^2 \mu)]}{(\sigma_1^2 \mu)^{(1/\sigma^2)} \Gamma(1/\sigma^2)}

for y>0, \mu>0 where \sigma_1=\sigma \mu^{(\nu/2)-1} \sigma>0 and -\infty <\nu< \infty see pp. 442-443 of Rigby et al. (2019).

Value

GAF() returns a gamlss.family object which can be used to fit the gamma family in the gamlss() function.

Note

For the function GAF(), \mu is the mean and \sigma \mu^{\nu/2} is the standard deviation of the gamma family. The GAF is design for fitting regression type models where the variance is proportional to a power of the mean.

Note that because the high correlation between the sigma and the nu parameter the mixed() method should be used in the fitting.

Author(s)

Mikis Stasinopoulos, Robert Rigby and Fernanda De Bastiani

References

Enki, D G, Noufaily, A., Farrington, P., Garthwaite, P., Andrews, N. and Charlett, A. (2017) Taylor's power law and the statistical modelling of infectious disease surveillance data, Journal of the Royal Statistical Society: Series A (Statistics in Society), volume=180, number=1, pages=45-72.

Rigby, R. A. and Stasinopoulos D. M. (2005). Generalized additive models for location, scale and shape,(with discussion), Appl. Statist., 54, part 3, pp 507-554.

Rigby, R. A., Stasinopoulos, D. M., Heller, G. Z., and De Bastiani, F. (2019) Distributions for modeling location, scale, and shape: Using GAMLSS in R, Chapman and Hall/CRC, doi:10.1201/9780429298547. An older version can be found in https://www.gamlss.com/.

Stasinopoulos D. M. Rigby R.A. (2007) Generalized additive models for location scale and shape (GAMLSS) in R. Journal of Statistical Software, Vol. 23, Issue 7, Dec 2007, doi:10.18637/jss.v023.i07..

Stasinopoulos D. M., Rigby R.A., Heller G., Voudouris V., and De Bastiani F., (2017) Flexible Regression and Smoothing: Using GAMLSS in R, Chapman and Hall/CRC. doi:10.1201/b21973

(see also https://www.gamlss.com/).

See Also

gamlss.family, GA, GG

Examples

GAF()
## Not run: 
m1<-gamlss(y~poly(x,2),data=abdom,family=GAF, method=mixed(1,100),  
           c.crit=0.00001)
# using RS()
m2<-gamlss(y~poly(x,2),data=abdom,family=GAF,  n.cyc=5000, c.crit=0.00001)
# the estimates of nu slightly different
fitted(m1, "nu")[1]
fitted(m2, "nu")[1]
# global deviance almost identical
AIC(m1, m2)

## End(Not run)

Create a GAMLSS Distribution

Description

A single class and corresponding methods encompassing all distributions from the gamlss.dist package using the workflow from the distributions3 package.

Usage

GAMLSS(family, mu, sigma, tau, nu)

Arguments

Details

The S3 class GAMLSS provides a unified workflow based on the distributions3 package (see Zeileis et al. 2022) for all distributions from the gamlss.dist package. The idea is that from fitted gamlss model objects predicted probability distributions can be obtained for which moments (mean, variance, etc.), probabilities, quantiles, etc. can be obtained with corresponding generic functions.

The constructor function GAMLSS sets up a distribution object, representing a distribution from the GAMLSS (generalized additive model of location, scale, and shape) framework by the corresponding parameters plus a family attribute, e.g., NO for the normal distribution or BI for the binomial distribution. There can be up to four parameters, called mu (often some sort of location parameter), sigma (often some sort of scale parameter), tau and nu (other parameters, e.g., capturing shape, etc.).

All parameters can also be vectors, so that it is possible to define a vector of GAMLSS distributions from the same family with potentially different parameters. All parameters need to have the same length or must be scalars (i.e., of length 1) which are then recycled to the length of the other parameters.

Note that not all distributions use all four parameters, i.e., some use just a subset. In that case, the corresponding arguments in GAMLSS should be unspecified, NULL, or NA.

For the GAMLSS distribution objects there is a wide range of standard methods available to the generics provided in the distributions3 package: pdf and log_pdf for the (log-)density (PDF), cdf for the probability from the cumulative distribution function (CDF), quantile for quantiles, random for simulating random variables, and support for the support interval (minimum and maximum). Internally, these methods rely on the usual d/p/q/r functions provided in gamlss.dist, see the manual pages of the individual families. The methods is_discrete and is_continuous can be used to query whether the distributions are discrete on the entire support or continuous on the entire support, respectively.

See the examples below for an illustration of the workflow for the class and methods.

Value

A GAMLSS object, inheriting from distribution.

References

Zeileis A, Lang MN, Hayes A (2022). “distributions3: From Basic Probability to Probabilistic Regression.” Presented at useR! 2022 - The R User Conference. Slides, video, vignette, code at https://www.zeileis.org/news/user2022/.

See Also

gamlss.family

Examples

## package and random seed
library("distributions3")
set.seed(6020)

## three Weibull distributions
X <- GAMLSS("WEI", mu = c(1, 1, 2), sigma = c(1, 2, 2))
X

## moments
mean(X)
variance(X)

## support interval (minimum and maximum)
support(X)
is_discrete(X)
is_continuous(X)

## simulate random variables
random(X, 5)

## histograms of 1,000 simulated observations
x <- random(X, 1000)
hist(x[1, ], main = "WEI(1,1)")
hist(x[2, ], main = "WEI(1,2)")
hist(x[3, ], main = "WEI(2,2)")

## probability density function (PDF) and log-density (or log-likelihood)
x <- c(2, 2, 1)
pdf(X, x)
pdf(X, x, log = TRUE)
log_pdf(X, x)

## cumulative distribution function (CDF)
cdf(X, x)

## quantiles
quantile(X, 0.5)

## cdf() and quantile() are inverses
cdf(X, quantile(X, 0.5))
quantile(X, cdf(X, 1))

## all methods above can either be applied elementwise or for
## all combinations of X and x, if length(X) = length(x),
## also the result can be assured to be a matrix via drop = FALSE
p <- c(0.05, 0.5, 0.95)
quantile(X, p, elementwise = FALSE)
quantile(X, p, elementwise = TRUE)
quantile(X, p, elementwise = TRUE, drop = FALSE)

## compare theoretical and empirical mean from 1,000 simulated observations
cbind(
  "theoretical" = mean(X),
  "empirical" = rowMeans(random(X, 1000))
)

The generalized Beta type 1 distribution for fitting a GAMLSS

Description

This function defines the generalized beta type 1 distribution, a four parameter distribution. The function GB1 creates a gamlss.family object which can be used to fit the distribution using the function gamlss(). Note the range of the response variable is from zero to one. The functions dGB1, GB1, qGB1 and rGB1 define the density, distribution function, quantile function and random generation for the generalized beta type 1 distribution.

Usage

GB1(mu.link = "logit", sigma.link = "logit", nu.link = "log", 
      tau.link = "log")
dGB1(x, mu = 0.5, sigma = 0.4, nu = 1, tau = 1, log = FALSE)
pGB1(q, mu = 0.5, sigma = 0.4, nu = 1, tau = 1, lower.tail = TRUE, 
       log.p = FALSE)
qGB1(p, mu = 0.5, sigma = 0.4, nu = 1, tau = 1, lower.tail = TRUE, 
       log.p = FALSE)
rGB1(n, mu = 0.5, sigma = 0.4, nu = 1, tau = 1)

Arguments

Details

The probability density function of the Generalized Beta type 1, (GB1), is defined as

f(y|\mu,\sigma\,\nu,\tau)= \frac{\tau \nu^\beta y^{\tau \alpha-1} \left( 1-y^\tau\right)^{\beta-1}}{B(\alpha,\beta)[\nu+(1-\nu) y^\tau]^{\alpha+\beta}}

where 0 < y < 1 , 0<\mu<1, 0<\sigma<1, \nu>0, \tau>0 and where \alpha = \mu(1-\sigma^2)/\sigma^2 and \beta=(1-\mu)(1-\sigma^2)/\sigma^2, and \alpha>0, \beta>0. Note the \mu=\alpha /(\alpha+\beta), \sigma = (\alpha+\beta+1)^{-1/2} see pp. 464-465 of Rigby et al. (2019).

Value

GB1() returns a gamlss.family object which can be used to fit the GB1 distribution in the gamlss() function. dGB1() gives the density, pGB1() gives the distribution function, qGB1() gives the quantile function, and rGB1() generates random deviates.

Warning

The qSHASH and rSHASH are slow since they are relying on golden section for finding the quantiles

Author(s)

Bob Rigby and Mikis Stasinopoulos

References

Rigby, R. A. and Stasinopoulos D. M. (2005). Generalized additive models for location, scale and shape,(with discussion), Appl. Statist., 54, part 3, pp 507-554.

Rigby, R. A., Stasinopoulos, D. M., Heller, G. Z., and De Bastiani, F. (2019) Distributions for modeling location, scale, and shape: Using GAMLSS in R, Chapman and Hall/CRC,doi:10.1201/9780429298547. An older version can be found in https://www.gamlss.com/.

Stasinopoulos D. M. Rigby R.A. (2007) Generalized additive models for location scale and shape (GAMLSS) in R. Journal of Statistical Software, Vol. 23, Issue 7, Dec 2007, doi:10.18637/jss.v023.i07.

Stasinopoulos D. M., Rigby R.A., Heller G., Voudouris V., and De Bastiani F., (2017) Flexible Regression and Smoothing: Using GAMLSS in R, Chapman and Hall/CRC. doi:10.1201/b21973

(see also https://www.gamlss.com/).

See Also

gamlss.family, JSU, BCT

Examples

GB1()   # 
y<- rGB1(200, mu=.1, sigma=.6, nu=1, tau=4)
hist(y)
# library(gamlss)
# histDist(y, family=GB1, n.cyc=60)
curve(dGB1(x, mu=.1 ,sigma=.6, nu=1, tau=4), 0.01, 0.99, main = "The GB1  
           density mu=0.1, sigma=.6, nu=1, tau=4")

The generalized Beta type 2 and generalized Pareto distributions for fitting a GAMLSS

Description

This function defines the generalized beta type 2 distribution, a four parameter distribution. The function GB2 creates a gamlss.family object which can be used to fit the distribution using the function gamlss(). The response variable is in the range from zero to infinity. The functions dGB2, GB2, qGB2 and rGB2 define the density, distribution function, quantile function and random generation for the generalized beta type 2 distribution. The generalised Pareto GP distribution is defined by setting the parameters sigma and nu of the GB2 distribution to 1.

Usage

GB2(mu.link = "log", sigma.link = "log", nu.link = "log", 
     tau.link = "log")
dGB2(x, mu = 1, sigma = 1, nu = 1, tau = 0.5, log = FALSE)
pGB2(q, mu = 1, sigma = 1, nu = 1, tau = 0.5, lower.tail = TRUE, 
     log.p = FALSE)
qGB2(p, mu = 1, sigma = 1, nu = 1, tau = 0.5, lower.tail = TRUE, 
     log.p = FALSE)
rGB2(n, mu = 1, sigma = 1, nu = 1, tau = 0.5)

GP(mu.link = "log", sigma.link = "log")
dGP(x, mu = 1, sigma = 1, log = FALSE)
pGP(q, mu = 1, sigma = 1, lower.tail = TRUE, log.p = FALSE)
qGP(p, mu = 1, sigma = 1, lower.tail = TRUE, log.p = FALSE)
rGP(n, mu = 1, sigma = 1)

Arguments

Details

The probability density function of the Generalized Beta type 2, (GB2), is defined as

f(y|\mu,\sigma,\nu,\tau)= |\sigma| y^{\hspace{0.01cm}\sigma v-1 } \{\mu^{\sigma \nu} \hspace{0.05cm}B(\nu,\tau) [1+(y/\mu)^\sigma]^{\nu+\tau}\}^{-1}

where y > 0, \mu>0, \sigma >0, \nu>0 and \tau>0 see pp. 452-453 of Rigby et al. (2019).

Note that by setting \sigma=1 we have the Pearson type VI, by setting \nu=1 we have the Burr type XII and by setting \tau=1 the Burr type III.

Value

GB2() returns a gamlss.family object which can be used to fit the GB2 distribution in the gamlss() function. dGB2() gives the density, pGB2() gives the distribution function, qGB2() gives the quantile function, and rGB2() generates random deviates.

Warning

The qSHASH and rSHASH are slow since they are relying on golden section for finding the quantiles

Author(s)

Bob Rigby and Mikis Stasinopoulos

References

Rigby, R. A. and Stasinopoulos D. M. (2005). Generalized additive models for location, scale and shape, (with discussion), Appl. Statist., 54, part 3, pp 507-554.

Rigby, R. A., Stasinopoulos, D. M., Heller, G. Z., and De Bastiani, F. (2019) Distributions for modeling location, scale, and shape: Using GAMLSS in R, Chapman and Hall/CRC, doi:10.1201/9780429298547. An older version can be found in https://www.gamlss.com/.

Stasinopoulos D. M. Rigby R.A. (2007) Generalized additive models for location scale and shape (GAMLSS) in R. Journal of Statistical Software, Vol. 23, Issue 7, Dec 2007, doi:10.18637/jss.v023.i07.

Stasinopoulos D. M., Rigby R.A., Heller G., Voudouris V., and De Bastiani F., (2017) Flexible Regression and Smoothing: Using GAMLSS in R, Chapman and Hall/CRC. doi:10.1201/b21973

(see also https://www.gamlss.com/).

See Also

gamlss.family, JSU, BCT

Examples

GB2()   # 
y<- rGB2(200, mu=5, sigma=2, nu=1, tau=1)
library(MASS)
truehist(y)
fx<-dGB2(seq(0.01, 20, length=200), mu=5 ,sigma=2, nu=1, tau=1)
lines(seq(0.01,20,length=200),fx)
integrate(function(x) x*dGB2(x=x, mu=5, sigma=2, nu=1, tau=1), 0, Inf)
mean(y)
curve(dGB2(x, mu=5 ,sigma=2, nu=1, tau=1), 0.01, 20, 
            main = "The GB2  density mu=5, sigma=2, nu=1, tau=4")

Geometric distribution for fitting a GAMLSS model

Description

The functions GEOMo() and GEOM() define two parametrizations of the geometric distribution. The geometric distribution is a one parameter distribution, for a gamlss.family object to be used in GAMLSS fitting using the function gamlss(). The mean of GEOM() is equal to the parameter mu. The functions dGEOM, pGEOM, qGEOM and rGEOM define the density, distribution function, quantile function and random generation for the GEOM parameterization of the Geometric distribution.

Usage

GEOM(mu.link = "log")
dGEOM(x, mu = 2, log = FALSE)
pGEOM(q, mu = 2, lower.tail = TRUE, log.p = FALSE)
qGEOM(p, mu = 2, lower.tail = TRUE, log.p = FALSE)
rGEOM(n, mu = 2)
GEOMo(mu.link = "logit")
dGEOMo(x, mu = 0.5, log = FALSE)
pGEOMo(q, mu = 0.5, lower.tail = TRUE, log.p = FALSE)
qGEOMo(p, mu = 0.5, lower.tail = TRUE, log.p = FALSE)
rGEOMo(n, mu = 0.5)

Arguments

Details

The parameterization of the original geometric distribution in the function GEOM is:

f(y|\mu) = \mu^y/(\mu+1)^{y+1}

for y=0,1,2,3,... and \mu>0, see pp 472-473 of Rigby et al. (2019).

The parameterization of the original geometric distribution, GEOMo is

f(y|\mu) = (1-\mu)^y\, \mu

for eqny=0,1,2,3,... and \mu>0, see pp 473-474 of Rigby et al. (2019).

Value

returns a gamlss.family object which can be used to fit a Geometric distribution in the gamlss() function.

Author(s)

Fiona McElduff, Bob Rigby and Mikis Stasinopoulos.

References

Johnson, N. L., Kemp, A. W., and Kotz, S. (2005). Univariate discrete distributions. Wiley.

Rigby, R. A. and Stasinopoulos D. M. (2005). Generalized additive models for location, scale and shape,(with discussion), Appl. Statist., 54, part 3, pp 507-554.

Rigby, R. A., Stasinopoulos, D. M., Heller, G. Z., and De Bastiani, F. (2019) Distributions for modeling location, scale, and shape: Using GAMLSS in R, Chapman and Hall/CRC, doi:10.1201/9780429298547. An older version can be found in https://www.gamlss.com/.

Stasinopoulos D. M. Rigby R.A. (2007) Generalized additive models for location scale and shape (GAMLSS) in R. Journal of Statistical Software, Vol. 23, Issue 7, Dec 2007, doi:10.18637/jss.v023.i07.

Stasinopoulos D. M., Rigby R.A., Heller G., Voudouris V., and De Bastiani F., (2017) Flexible Regression and Smoothing: Using GAMLSS in R, Chapman and Hall/CRC. doi:10.1201/b21973

(see also https://www.gamlss.com/).

See Also

gamlss.family

Examples

par(mfrow=c(2,2))
y<-seq(0,20,1)
plot(y, dGEOM(y), type="h")
q <- seq(0, 20, 1)
plot(q, pGEOM(q), type="h")
p<-seq(0.0001,0.999,0.05)
plot(p , qGEOM(p), type="s")
dat <- rGEOM(100)
hist(dat)
#summary(gamlss(dat~1, family=GEOM))
par(mfrow=c(2,2))
y<-seq(0,20,1)
plot(y, dGEOMo(y), type="h")
q <- seq(0, 20, 1)
plot(q, pGEOMo(q), type="h")
p<-seq(0.0001,0.999,0.05)
plot(p , qGEOMo(p), type="s")
dat <- rGEOMo(100)
hist(dat)
#summary(gamlss(dat~1, family="GE"))

Generalized Gamma distribution for fitting a GAMLSS

Description

The function GG defines the generalized gamma distribution, a three parameter distribution, for a gamlss.family object to be used in GAMLSS fitting using the function gamlss(). The parameterization used has the mean of the distribution equal to mu and the variance equal to (\sigma^2)(\mu^2). The functions dGG, pGG, qGG and rGG define the density, distribution function, quantile function and random generation for the specific parameterization of the generalized gamma distribution defined by function GG.

Usage

GG(mu.link = "log", sigma.link = "log", 
                       nu.link = "identity")
dGG(x, mu=1, sigma=0.5, nu=1,  
                      log = FALSE)
pGG(q, mu=1, sigma=0.5, nu=1,  lower.tail = TRUE, 
                     log.p = FALSE)
qGG(p, mu=1, sigma=0.5, nu=1,  lower.tail = TRUE, 
                     log.p = FALSE )
rGG(n, mu=1, sigma=0.5, nu=1)

Arguments

Details

The specific parameterization of the generalized gamma distribution used in GG is

f(y|\mu,\sigma,\nu)= \frac{\theta^\theta z^\theta |\nu| e^(-\theta z)}{(\Gamma(\theta)y)}

where z =(y/\mu)^\nu, \theta = 1/(\sigma^2|\nu|^2) for y>0, \mu>0, \sigma>0 and -\infty<\nu<+\infty see pp. 443-444 of Rigby et al. (2019). Note that for \nu=0 the distribution is log normal.

Value

GG() returns a gamlss.family object which can be used to fit a generalized gamma distribution in the gamlss() function. dGG() gives the density, pGG() gives the distribution function, qGG() gives the quantile function, and rGG() generates random deviates.

Author(s)

Mikis Stasinopoulos, Bob Rigby and Nicoleta Motpan

References

Lopatatzidis, A. and Green, P. J. (2000), Nonparametric quantile regression using the gamma distribution, unpublished.

Rigby, R. A. and Stasinopoulos D. M. (2005). Generalized additive models for location, scale and shape,(with discussion), Appl. Statist., 54, part 3, pp 507-554.

Rigby, R. A., Stasinopoulos, D. M., Heller, G. Z., and De Bastiani, F. (2019) Distributions for modeling location, scale, and shape: Using GAMLSS in R, Chapman and Hall/CRC, doi:10.1201/9780429298547. An older version can be found in https://www.gamlss.com/.

Stasinopoulos D. M. Rigby R.A. (2007) Generalized additive models for location scale and shape (GAMLSS) in R. Journal of Statistical Software, Vol. 23, Issue 7, Dec 2007, doi:10.18637/jss.v023.i07.

Stasinopoulos D. M., Rigby R.A., Heller G., Voudouris V., and De Bastiani F., (2017) Flexible Regression and Smoothing: Using GAMLSS in R, Chapman and Hall/CRC. doi:10.1201/b21973

(see also https://www.gamlss.com/).

See Also

gamlss.family, GA

Examples

y<-rGG(100,mu=1,sigma=0.1, nu=-.5) # generates 100 random observations  
hist(y)
# library(gamlss)
#histDist(y, family=GG)
#m1 <-gamlss(y~1,family=GG)
#prof.dev(m1, "nu", min=-2, max=2, step=0.2)

Generalized Inverse Gaussian distribution for fitting a GAMLSS

Description

The function GIG defines the generalized inverse gaussian distribution, a three parameter distribution, for a gamlss.family object to be used in GAMLSS fitting using the function gamlss(). The functions DIG, pGIG, GIG and rGIG define the density, distribution function, quantile function and random generation for the specific parameterization of the generalized inverse gaussian distribution defined by function GIG.

Usage

GIG(mu.link = "log", sigma.link = "log", 
                       nu.link = "identity")
dGIG(x, mu=1, sigma=1, nu=1,  
                      log = FALSE)
pGIG(q, mu=1, sigma=1, nu=1,  lower.tail = TRUE, 
                     log.p = FALSE)
qGIG(p, mu=1, sigma=1, nu=1,  lower.tail = TRUE, 
                     log.p = FALSE)
rGIG(n, mu=1, sigma=1, nu=1, ...)

Arguments

Details

The specific parameterization of the generalized inverse gaussian distribution used in GIG is

f(y|\mu,\sigma,\nu)= (\frac{b}{\mu})^\nu \left[ \frac{y^{\nu-1}}{2 K_{\nu}(\sigma^{-2}) }\right] \exp \left[-\frac{1}{2\sigma^2} \left(\frac{b y }{\mu}+\frac{\mu}{ b y} \right)\right]

where b = \frac{K_{\nu+1}(\frac{1}{\sigma^2})}{K_{\nu}(\frac{1}{\sigma^{-2}})}, for y>0, \mu>0, \sigma>0 and -\infty<\nu<+\infty see pp 445-446 of Rigby et al. (2019).

Value

GIG() returns a gamlss.family object which can be used to fit a generalized inverse gaussian distribution in the gamlss() function. DIG() gives the density, pGIG() gives the distribution function, GIG() gives the quantile function, and rGIG() generates random deviates.

Author(s)

Mikis Stasinopoulos, Bob Rigby and Nicoleta Motpan

References

Rigby, R. A. and Stasinopoulos D. M. (2005). Generalized additive models for location, scale and shape,(with discussion), Appl. Statist., 54, part 3, pp 507-554.

Rigby, R. A., Stasinopoulos, D. M., Heller, G. Z., and De Bastiani, F. (2019) Distributions for modeling location, scale, and shape: Using GAMLSS in R, Chapman and Hall/CRC, doi:10.1201/9780429298547. An older version can be found in https://www.gamlss.com/.

Stasinopoulos D. M. Rigby R.A. (2007) Generalized additive models for location scale and shape (GAMLSS) in R. Journal of Statistical Software, Vol. 23, Issue 7, Dec 2007, doi:10.18637/jss.v023.i07.

Stasinopoulos D. M., Rigby R.A., Heller G., Voudouris V., and De Bastiani F., (2017) Flexible Regression and Smoothing: Using GAMLSS in R, Chapman and Hall/CRC. doi:10.1201/b21973

Jorgensen B. (1982) Statistical properties of the generalized inverse Gaussian distribution, Series: Lecture notes in statistics; 9, New York : Springer-Verlag.

(see also https://www.gamlss.com/).

See Also

gamlss.family, IG

Examples

y<-rGIG(100,mu=1,sigma=1, nu=-0.5) # generates 1000 random observations 
hist(y)
# library(gamlss)
# histDist(y, family=GIG) 

The generalised Poisson distribution

Description

The GPO() function defines the generalised Poisson distribution, a two parameter discrete distribution, for a gamlss.family object to be used in GAMLSS fitting using the function gamlss(). The functions dGPO, pGPO, qGPO and rGPO define the density, distribution function, quantile function and random generation for the Delaporte GPO(), distribution.

Usage

GPO(mu.link = "log", sigma.link = "log")

dGPO(x, mu = 1, sigma = 1, log = FALSE)

pGPO(q, mu = 1, sigma = 1, lower.tail = TRUE, log.p = FALSE)

qGPO(p, mu = 1, sigma = 1, lower.tail = TRUE, log.p = FALSE, max.value = 10000)

rGPO(n, mu = 1, sigma = 1,  max.value = 10000)

Arguments

Details

The probability function of the Generalised Poisson distribution is given by

P(Y=y|\mu,\sigma)= \left(\frac{\mu}{1+\sigma \mu}\right)^y \frac{\left(1+\sigma y \right)^{y-1}}{y!} \exp \left[\frac{- \mu \left( 1+\sigma y\right)}{1+\sigma \mu} \right]

for y=0,1,2,...,\infty where \mu>0 and \sigma>0 see pp. 481-483 of Rigby et al. (2019).

Value

Returns a gamlss.family object which can be used to fit a Generalised Poisson distribution in the gamlss() function.

Author(s)

Rigby, R. A., Stasinopoulos D. M.

References

Rigby, R. A. and Stasinopoulos D. M. (2005). Generalized additive models for location, scale and shape,(with discussion), Appl. Statist., 54, part 3, pp 507-554.

Rigby, R. A., Stasinopoulos, D. M., Heller, G. Z., and De Bastiani, F. (2019) Distributions for modeling location, scale, and shape: Using GAMLSS in R, Chapman and Hall/CRC, doi:10.1201/9780429298547. An older version can be found in https://www.gamlss.com/.

Stasinopoulos D. M. Rigby R.A. (2007) Generalized additive models for location scale and shape (GAMLSS) in R. Journal of Statistical Software, Vol. 23, Issue 7, Dec 2007, doi:10.18637/jss.v023.i07.

Stasinopoulos D. M., Rigby R.A., Heller G., Voudouris V., and De Bastiani F., (2017) Flexible Regression and Smoothing: Using GAMLSS in R, Chapman and Hall/CRC. doi:10.1201/b21973

(see also https://www.gamlss.com/).

See Also

gamlss.family, PO , DPO

Examples

GPO()# gives information about the default links for the
#plot the pdf using plot 
plot(function(y) dGPO(y, mu=10, sigma=1 ), from=0, to=100, n=100+1, type="h") # pdf
# plot the cdf
plot(seq(from=0,to=100),pGPO(seq(from=0,to=100), mu=10, sigma=1), type="h")   # cdf
# generate random sample
tN <- table(Ni <- rGPO(100, mu=5, sigma=1))
r <- barplot(tN, col='lightblue')

The generalized t distribution for fitting a GAMLSS

Description

This function defines the generalized t distribution, a four parameter distribution, for a gamlss.family object to be used for a GAMLSS fitting using the function gamlss(). The functions dGT, pGT, qGT and rGT define the density, distribution function, quantile function and random generation for the generalized t distribution.

Usage

GT(mu.link = "identity", sigma.link = "log", nu.link = "log", 
   tau.link = "log")
dGT(x, mu = 0, sigma = 1, nu = 3, tau = 1.5, log = FALSE)
pGT(q, mu = 0, sigma = 1, nu = 3, tau = 1.5, lower.tail = TRUE, 
   log.p = FALSE)
qGT(p, mu = 0, sigma = 1, nu = 3, tau = 1.5, lower.tail = TRUE, 
   log.p = FALSE)
rGT(n, mu = 0, sigma = 1, nu = 3, tau = 1.5)

Arguments

Details

The probability density function of the generalized t distribution, (GT), , is defined as

f(y|\mu,\sigma\,\nu,\tau)= \tau \left\{2\sigma \nu^{1/\tau} B\left(\frac{1}{\tau},\nu\right)[1+|z|^{\tau}/\nu]^{\nu+1/\tau} \right\}^{-1}

where -\infty < y < \infty , z=(y-\mu)/\sigma \mu=(-\infty,+\infty), \sigma>0, \nu>0 and \tau>0, see pp. 387-388 of Rigby et al. (2019).

Value

GT() returns a gamlss.family object which can be used to fit the GT distribution in the gamlss() function. dGT() gives the density, pGT() gives the distribution function, qGT() gives the quantile function, and rGT() generates random deviates.

Warning

The qGT and rGT are slow since they are relying on optimization for finding the quantiles

Author(s)

Bob Rigby and Mikis Stasinopoulos

References

Rigby, R. A. and Stasinopoulos D. M. (2005). Generalized additive models for location, scale and shape,(with discussion), Appl. Statist., 54, part 3, pp 507-554.

Rigby, R. A., Stasinopoulos, D. M., Heller, G. Z., and De Bastiani, F. (2019) Distributions for modeling location, scale, and shape: Using GAMLSS in R, Chapman and Hall/CRC, doi:10.1201/9780429298547. An older version can be found in https://www.gamlss.com/.

Stasinopoulos D. M. Rigby R.A. (2007) Generalized additive models for location scale and shape (GAMLSS) in R. Journal of Statistical Software, Vol. 23, Issue 7, Dec 2007, doi:10.18637/jss.v023.i07.

(see also https://www.gamlss.com/).

See Also

gamlss.family, JSU, BCT

Examples

GT()   # 
y<- rGT(200, mu=5, sigma=1, nu=1, tau=4)
hist(y)
curve(dGT(x, mu=5 ,sigma=2,nu=1, tau=4), -2, 11, 
      main = "The GT  density mu=5 ,sigma=1, nu=1, tau=4")
# library(gamlss)
# m1<-gamlss(y~1, family=GT) 

The Gumbel distribution for fitting a GAMLSS

Description

The function GU defines the Gumbel distribution, a two parameter distribution, for a gamlss.family object to be used in GAMLSS fitting using the function gamlss(). The functions dGU, pGU, qGU and rGU define the density, distribution function, quantile function and random generation for the specific parameterization of the Gumbel distribution.

Usage

GU(mu.link = "identity", sigma.link = "log")
dGU(x, mu = 0, sigma = 1, log = FALSE)
pGU(q, mu = 0, sigma = 1, lower.tail = TRUE, log.p = FALSE)
qGU(p, mu = 0, sigma = 1, lower.tail = TRUE, log.p = FALSE)
rGU(n, mu = 0, sigma = 1)

Arguments

Details

The specific parameterization of the Gumbel distribution used in GU is

f(y|\mu,\sigma)=\frac{1}{\sigma} \hspace{1mm} \exp\left\{\left(\frac{y-\mu}{\sigma}\right)-\exp\left(\frac{y-\mu}{\sigma}\right)\right\}

for y=(-\infty,\infty), \mu=(-\infty,+\infty) and \sigma>0, see pp. 366-367 of Rigby et al. (2019).

Value

GU() returns a gamlss.family object which can be used to fit a Gumbel distribution in the gamlss() function. dGU() gives the density, pGU() gives the distribution function, qGU() gives the quantile function, and rGU() generates random deviates.

Note

The mean of the distribution is \mu-0.57722 \sigma and the variance is \pi^2 \sigma^2/6.

Author(s)

Mikis Stasinopoulos, Bob Rigby and Calliope Akantziliotou

References

Rigby, R. A. and Stasinopoulos D. M. (2005). Generalized additive models for location, scale and shape,(with discussion), Appl. Statist., 54, part 3, pp 507-554.

Rigby, R. A., Stasinopoulos, D. M., Heller, G. Z., and De Bastiani, F. (2019) Distributions for modeling location, scale, and shape: Using GAMLSS in R, Chapman and Hall/CRC, doi:10.1201/9780429298547. An older version can be found in https://www.gamlss.com/.

Stasinopoulos D. M. Rigby R.A. (2007) Generalized additive models for location scale and shape (GAMLSS) in R. Journal of Statistical Software, Vol. 23, Issue 7, Dec 2007, \doi10.18637/jss.v023.i07.

Stasinopoulos D. M., Rigby R.A., Heller G., Voudouris V., and De Bastiani F., (2017) Flexible Regression and Smoothing: Using GAMLSS in R, Chapman and Hall/CRC. doi:10.1201/b21973

(see also https://www.gamlss.com/).

See Also

gamlss.family, RG

Examples

plot(function(x) dGU(x, mu=0,sigma=1), -6, 3, 
 main = "{Gumbel  density mu=0,sigma=1}")
GU()# gives information about the default links for the Gumbel distribution      
dat<-rGU(100, mu=10, sigma=2) # generates 100 random observations 
hist(dat)
# library(gamlss)
# gamlss(dat~1,family=GU) # fits a constant for each parameter mu and sigma 

Inverse Gaussian distribution for fitting a GAMLSS

Description

The function IG(), or equivalently Inverse.Gaussian(), defines the inverse Gaussian distribution, a two parameter distribution, for a gamlss.family object to be used in GAMLSS fitting using the function gamlss(). The functions dIG, pIG, qIG and rIG define the density, distribution function, quantile function and random generation for the specific parameterization of the Inverse Gaussian distribution defined by function IG.

Usage

IG(mu.link = "log", sigma.link = "log")
dIG(x, mu = 1, sigma = 1, log = FALSE)
pIG(q, mu = 1, sigma = 1, lower.tail = TRUE, log.p = FALSE)
qIG(p, mu = 1, sigma = 1, lower.tail = TRUE, log.p = FALSE)
rIG(n, mu = 1, sigma = 1, ...)

Arguments

Details

Definition file for inverse Gaussian distribution.

f(y|\mu,\sigma)= \frac{1}{\sqrt{2 \pi \sigma^2 y^3}} \hspace{1mm} \exp\left\{-\frac{1}{2 \mu^2 \sigma^2 y}\hspace{1mm}(y-\mu)^2\right\}

for y>0, \mu>0 and \sigma>0 see pp. 426-427 of Rigby et al. (2019).

Value

returns a gamlss.family object which can be used to fit a inverse Gaussian distribution in the gamlss() function.

Note

\mu is the mean and \sigma^2 \mu^3 is the variance of the inverse Gaussian

Author(s)

Mikis Stasinopoulos, Bob Rigby and Calliope Akantziliotou

References

Rigby, R. A. and Stasinopoulos D. M. (2005). Generalized additive models for location, scale and shape,(with discussion), Appl. Statist., 54, part 3, pp 507-554.

Rigby, R. A., Stasinopoulos, D. M., Heller, G. Z., and De Bastiani, F. (2019) Distributions for modeling location, scale, and shape: Using GAMLSS in R, Chapman and Hall/CRC, doi:10.1201/9780429298547. An older version can be found in https://www.gamlss.com/.

Stasinopoulos D. M. Rigby R.A. (2007) Generalized additive models for location scale and shape (GAMLSS) in R. Journal of Statistical Software, Vol. 23, Issue 7, Dec 2007, \doi10.18637/jss.v023.i07.

Stasinopoulos D. M., Rigby R.A., Heller G., Voudouris V., and De Bastiani F., (2017) Flexible Regression and Smoothing: Using GAMLSS in R, Chapman and Hall/CRC. doi:10.1201/b21973

(see also https://www.gamlss.com/).

See Also

gamlss.family , GA, GIG

Examples

IG()# gives information about the default links for the normal distribution
# library(gamlss)
# data(rent)        
# gamlss(R~cs(Fl),family=IG, data=rent) # 
plot(function(x)dIG(x, mu=1,sigma=.5), 0.01, 6, 
 main = "{Inverse Gaussian  density mu=1,sigma=0.5}")
plot(function(x)pIG(x, mu=1,sigma=.5), 0.01, 6, 
 main = "{Inverse Gaussian  cdf mu=1,sigma=0.5}")

Inverse Gamma distribution for fitting a GAMLSS

Description

The function IGAMMA() defines the Inverse Gamma distribution, a two parameter distribution, for a gamlss.family object to be used in GAMLSS fitting using the function gamlss(), with parameters mu (the mode) and sigma. The functions dIGAMMA, pIGAMMA, qIGAMMA and rIGAMMA define the density, distribution function, quantile function and random generation for the IGAMMA parameterization of the Inverse Gamma distribution.

Usage

IGAMMA(mu.link = "log", sigma.link="log")
dIGAMMA(x, mu = 1, sigma = .5, log = FALSE)
pIGAMMA(q, mu = 1, sigma = .5, lower.tail = TRUE, log.p = FALSE)
qIGAMMA(p, mu = 1, sigma = .5, lower.tail = TRUE, log.p = FALSE)
rIGAMMA(n, mu = 1, sigma = .5)

Arguments

Details

The parameterization of the Inverse Gamma distribution in the function IGAMMA is

f(y|\mu, \sigma) = \frac{\left[\mu\,(\alpha+1)\right]^{\alpha}}{\Gamma(\alpha)} \,y^{-(\alpha+1)}\, \exp{\left[-\frac{\mu\,(\alpha+1)}{y}\right]}

where \alpha = 1/(\sigma^2) for y>0, \mu>0 and \sigma>0 see pp. 424-426 of Rigby et al. (2019).

Value

returns a gamlss.family object which can be used to fit an Inverse Gamma distribution in the gamlss() function.

Note

For the function IGAMMA(), mu is the mode of the Inverse Gamma distribution.

Author(s)

Fiona McElduff, Bob Rigby and Mikis Stasinopoulos.

References

Rigby, R. A. and Stasinopoulos D. M. (2005). Generalized additive models for location, scale and shape,(with discussion), Appl. Statist., 54, part 3, pp 507-554.

Rigby, R. A., Stasinopoulos, D. M., Heller, G. Z., and De Bastiani, F. (2019) Distributions for modeling location, scale, and shape: Using GAMLSS in R, Chapman and Hall/CRC, doi:10.1201/9780429298547. An older version can be found in https://www.gamlss.com/.

Stasinopoulos D. M. Rigby R.A. (2007) Generalized additive models for location scale and shape (GAMLSS) in R. Journal of Statistical Software, Vol. 23, Issue 7, Dec 2007, doi:10.18637/jss.v023.i07.

Stasinopoulos D. M., Rigby R.A., Heller G., Voudouris V., and De Bastiani F., (2017) Flexible Regression and Smoothing: Using GAMLSS in R, Chapman and Hall/CRC. doi:10.1201/b21973

(see also https://www.gamlss.com/).

See Also

gamlss.family, GA

Examples

par(mfrow=c(2,2))
y<-seq(0.2,20,0.2)
plot(y, dIGAMMA(y), type="l")
q <- seq(0.2, 20, 0.2)
plot(q, pIGAMMA(q), type="l")
p<-seq(0.0001,0.999,0.05)
plot(p , qIGAMMA(p), type="l")
dat <- rIGAMMA(50)
hist(dat)
#summary(gamlss(dat~1, family="IGAMMA"))

The Johnson's Su distribution for fitting a GAMLSS

Description

This function defines the , a four parameter distribution, for a gamlss.family object to be used for a GAMLSS fitting using the function gamlss(). The functions dJSU, pJSU, qJSU and rJSU define the density, distribution function, quantile function and random generation for the the Johnson's Su distribution.

Usage

JSU(mu.link = "identity", sigma.link = "log", nu.link = "identity", tau.link = "log")
dJSU(x, mu = 0, sigma = 1, nu = 1, tau = 1, log = FALSE)
pJSU(q, mu = 0, sigma = 1, nu = 1, tau = 1, lower.tail = TRUE, log.p = FALSE)
qJSU(p, mu = 0, sigma = 1, nu = 1, tau = 1, lower.tail = TRUE, log.p = FALSE)
rJSU(n, mu = 0, sigma = 1, nu = 1, tau = 1)

Arguments

Details

The probability density function of the Jonhson's SU distribution, (JSU), is defined as

f(y|\mu,\sigma\,\nu,\tau)=\frac{\tau}{c \sigma(s^2+1)^{\frac{1}{2}}\sqrt{2\pi}} \hspace{1mm} \exp{\left[ -\frac{1}{2} z^2 \right]}

for -\infty < y < \infty , -\infty<\mu< \infty, \sigma>0, -\infty<\nu<\infty) and \tau>0 and where

z=-\nu+\tau \sinh^{-1}(s)

,

s = \frac{y-\mu+c \sigma w^{\frac{1}{2}} \sinh(\nu/\tau)}{c\sigma}

,

c = \left\{ \frac{1}{2} (w-1) \left[w \cosh(2\nu/\tau) +1) \right] \right\}^{-\frac{1}{2}}

and

w=e^{1/\tau^2}

see pp. 393-394 of Rigby et al. (2019).

This is a reparameterization of the original Johnson Su distribution, Johnson (1954), so the parameters mu and sigma are the mean and the standard deviation of the distribution. The parameter nu determines the skewness of the distribution with nu>0 indicating positive skewness and nu<0 negative. The parameter tau determines the kurtosis of the distribution. tau should be positive and most likely in the region from zero to 1. As tau goes to 0 (and for nu=0) the distribution approaches the the Normal density function. The distribution is appropriate for leptokurtic data that is data with kurtosis larger that the Normal distribution one.

Value

JSU() returns a gamlss.family object which can be used to fit a Johnson's Su distribution in the gamlss() function. dJSU() gives the density, pJSU() gives the distribution function, qJSU() gives the quantile function, and rJSU() generates random deviates.

Warning

The function JSU uses first derivatives square in the fitting procedure so standard errors should be interpreted with caution

Author(s)

Bob Rigby and Mikis Stasinopoulos

References

Johnson, N. L. (1954). Systems of frequency curves derived from the first law of Laplace., Trabajos de Estadistica, 5, 283-291.

Rigby, R. A. and Stasinopoulos D. M. (2005). Generalized additive models for location, scale and shape,(with discussion), Appl. Statist., 54, part 3, pp 507-554.

Rigby, R. A., Stasinopoulos, D. M., Heller, G. Z., and De Bastiani, F. (2019) Distributions for modeling location, scale, and shape: Using GAMLSS in R, Chapman and Hall/CRC, doi:10.1201/9780429298547. An older version can be found in https://www.gamlss.com/.

Stasinopoulos D. M. Rigby R. A. and Akantziliotou C. (2006) Instructions on how to use the GAMLSS package in R. Accompanying documentation in the current GAMLSS help files, (see also https://www.gamlss.com/).

Stasinopoulos D. M. Rigby R.A. (2007) Generalized additive models for location scale and shape (GAMLSS) in R. Journal of Statistical Software, Vol. 23, Issue 7, Dec 2007, doi:10.18637/jss.v023.i07.

Stasinopoulos D. M., Rigby R.A., Heller G., Voudouris V., and De Bastiani F., (2017) Flexible Regression and Smoothing: Using GAMLSS in R, Chapman and Hall/CRC. doi:10.1201/b21973

(see also https://www.gamlss.com/).

See Also

gamlss.family, JSUo, BCT

Examples

JSU()   
plot(function(x)dJSU(x, mu=0,sigma=1,nu=-1, tau=.5), -4, 4, 
 main = "The JSU  density mu=0,sigma=1,nu=-1, tau=.5")
plot(function(x) pJSU(x, mu=0,sigma=1,nu=-1, tau=.5), -4, 4, 
 main = "The JSU  cdf mu=0, sigma=1, nu=-1, tau=.5")
# library(gamlss)
# data(abdom) 
# h<-gamlss(y~cs(x,df=3), sigma.formula=~cs(x,1), family=JSU, data=abdom) 

The original Johnson's Su distribution for fitting a GAMLSS

Description

This function defines the , a four parameter distribution, for a gamlss.family object to be used for a GAMLSS fitting using the function gamlss(). The functions dJSUo, pJSUo, qJSUo and rJSUo define the density, distribution function, quantile function and random generation for the the Johnson's Su distribution.

Usage

JSUo(mu.link = "identity", sigma.link = "log", nu.link = "identity", tau.link = "log")
dJSUo(x, mu = 0, sigma = 1, nu = 0, tau = 1, log = FALSE)
pJSUo(q, mu = 0, sigma = 1, nu = 0, tau = 1, lower.tail = TRUE, log.p = FALSE)
qJSUo(p, mu = 0, sigma = 1, nu = 0, tau = 1, lower.tail = TRUE, log.p = FALSE)
rJSUo(n, mu = 0, sigma = 1, nu = 0, tau = 1)

Arguments

Details

The probability density function of the orininal Jonhson's SU distribution, (JSUo), is defined as

f(y|\mu,\sigma\,\nu,\tau)=\frac{\tau}{\sigma (z^{2}+1)^{\frac{1}{2}} \sqrt{2\pi}}\hspace{1mm} \exp{\left[ -\frac{1}{2} r^2 \right]}

for -\infty < y < \infty , \mu=(-\infty,+\infty), \sigma>0, \nu=(-\infty,+\infty) and \tau>0. where z = \frac{(y-\mu)}{\sigma}, r = \nu + \tau \sinh^{-1}(z), see pp. 389-390 of Rigby et al. (2019).

Value

JSUo() returns a gamlss.family object which can be used to fit a Johnson's Su distribution in the gamlss() function. dJSUo() gives the density, pJSUo() gives the distribution function, qJSUo() gives the quantile function, and rJSUo() generates random deviates.

Warning

The function JSU uses first derivatives square in the fitting procedure so standard errors should be interpreted with caution. It is recomented to be used only with method=mixed(2,20)

Author(s)

Mikis Stasinopoulos and Bob Rigby

References

Johnson, N. L. (1954). Systems of frequency curves derived from the first law of Laplace., Trabajos de Estadistica, 5, 283-291.

Rigby, R. A. and Stasinopoulos D. M. (2005). Generalized additive models for location, scale and shape,(with discussion), Appl. Statist., 54, part 3, pp 507-554.

Rigby, R. A., Stasinopoulos, D. M., Heller, G. Z., and De Bastiani, F. (2019) Distributions for modeling location, scale, and shape: Using GAMLSS in R, Chapman and Hall/CRC, doi:10.1201/9780429298547. An older version can be found in https://www.gamlss.com/.

Stasinopoulos D. M. Rigby R.A. (2007) Generalized additive models for location scale and shape (GAMLSS) in R. Journal of Statistical Software, Vol. 23, Issue 7, Dec 2007, doi:10.18637/jss.v023.i07.

Stasinopoulos D. M., Rigby R.A., Heller G., Voudouris V., and De Bastiani F., (2017) Flexible Regression and Smoothing: Using GAMLSS in R, Chapman and Hall/CRC. doi:10.1201/b21973

(see also https://www.gamlss.com/).

See Also

gamlss.family, JSU, BCT

Examples

JSU()   
plot(function(x)dJSUo(x, mu=0,sigma=1,nu=-1, tau=.5), -4, 15, 
 main = "The JSUo  density mu=0,sigma=1,nu=-1, tau=.5")
plot(function(x) pJSUo(x, mu=0,sigma=1,nu=-1, tau=.5), -4, 15, 
 main = "The JSUo  cdf mu=0, sigma=1, nu=-1, tau=.5")
# library(gamlss)
# data(abdom)
# h<-gamlss(y~cs(x,df=3), sigma.formula=~cs(x,1), family=JSUo, 
#          data=abdom, method=mixed(2,20)) 
# plot(h)

Logarithmic and zero adjusted logarithmic distributions for fitting a GAMLSS model

Description

The function LG defines the logarithmic distribution, a one parameter distribution, for a gamlss.family object to be used in GAMLSS fitting using the function gamlss(). The functions dLG, pLG, qLG and rLG define the density, distribution function, quantile function and random generation for the logarithmic , LG(), distribution.

The function ZALG defines the zero adjusted logarithmic distribution, a two parameter distribution, for a gamlss.family object to be used in GAMLSS fitting using the function gamlss(). The functions dZALG, pZALG, qZALG and rZALG define the density, distribution function, quantile function and random generation for the inflated logarithmic , ZALG(), distribution.

Usage

LG(mu.link = "logit")
dLG(x, mu = 0.5, log = FALSE)
pLG(q, mu = 0.5, lower.tail = TRUE, log.p = FALSE)
qLG(p, mu = 0.5, lower.tail = TRUE, log.p = FALSE, max.value = 10000)
rLG(n, mu = 0.5)
ZALG(mu.link = "logit", sigma.link = "logit")
dZALG(x, mu = 0.5, sigma = 0.1, log = FALSE)
pZALG(q, mu = 0.5, sigma = 0.1, lower.tail = TRUE, log.p = FALSE)
qZALG(p, mu = 0.5, sigma = 0.1, lower.tail = TRUE, log.p = FALSE)
rZALG(n, mu = 0.5, sigma = 0.1)

Arguments

Details

The parameterization of the logarithmic distribution in the function LG is

P(Y=y | \mu) = \alpha \mu^y / y

where for y=1,2,3,... with 0<\mu<1 and

\alpha = - [\log(1-\mu)]^{-1}.

see pp 474-475 of Rigby et al. (2019).

For the zero adjusted logarithmic distribution ZALG which is defined for y=0,1,2,3,... see pp 492-494 of Rigby et al. (2019).

Value

The function LG and ZALG return a gamlss.family object which can be used to fit a logarithmic and a zero inflated logarithmic distributions respectively in the gamlss() function.

Author(s)

Mikis Stasinopoulos, Bob Rigby

References

Johnson, Norman Lloyd; Kemp, Adrienne W; Kotz, Samuel (2005). "Chapter 7: Logarithmic and Lagrangian distributions". Univariate discrete distributions (3 ed.). John Wiley & Sons. ISBN 9780471272465.

Rigby, R. A. and Stasinopoulos D. M. (2005). Generalized additive models for location, scale and shape,(with discussion), Appl. Statist., 54, part 3, pp 507-554.

Rigby, R. A., Stasinopoulos, D. M., Heller, G. Z., and De Bastiani, F. (2019) Distributions for modeling location, scale, and shape: Using GAMLSS in R, Chapman and Hall/CRC, doi:10.1201/9780429298547. An older version can be found in https://www.gamlss.com/.

Stasinopoulos D. M. Rigby R.A. (2007) Generalized additive models for location scale and shape (GAMLSS) in R. Journal of Statistical Software, Vol. 23, Issue 7, Dec 2007, doi:10.18637/jss.v023.i07.

Rigby, R. A. and Stasinopoulos D. M. (2010) The gamlss.family distributions, (distributed with this package or see https://www.gamlss.com/)

Stasinopoulos D. M., Rigby R.A., Heller G., Voudouris V., and De Bastiani F., (2017) Flexible Regression and Smoothing: Using GAMLSS in R, Chapman and Hall/CRC. doi:10.1201/b21973

(see also https://www.gamlss.com/).

See Also

gamlss.family, PO, ZAP

Examples

LG() 
ZAP()
# creating data and plotting them 
 dat <- rLG(1000, mu=.3)
   r <- barplot(table(dat), col='lightblue')
dat1 <- rZALG(1000, mu=.3, sigma=.1)
  r1 <- barplot(table(dat1), col='lightblue')

Log Normal distribution for fitting in GAMLSS

Description

The functions LOGNO and LOGNO2 define a gamlss.family distribution to fits the log-Normal distribution. The difference between them is that while LOGNO retains the original parametrization for mu, (identical to the normal distribution NO) and therefore \mu=(-\infty,+\infty), the function LOGNO2 use mu as the median, so \mu=(0,+\infty).

The function LNO is more general and can fit a Box-Cox transformation to data using the gamlss() function. In the LOGNO (and LOGNO2) there are two parameters involved mu sigma, while in the LNO there are three parameters mu sigma, and the transformation parameter nu. The transformation parameter nu in LNO is a 'fixed' parameter (not estimated) and it has its default value equal to zero allowing the fitting of the log-normal distribution as in LOGNO. See the example below on how to fix nu to be a particular value. In order to estimate (or model) the parameter nu, use the gamlss.family BCCG distribution which uses a reparameterized version of the the Box-Cox transformation. The functions dLOGNO, pLOGNO, qLOGNO and rLOGNO define the density, distribution function, quantile function and random generation for the specific parameterization of the log-normal distribution.

The functions dLOGNO2, pLOGNO2, qLOGNO2 and rLOGNO2 define the density, distribution function, quantile function and random generation when mu is the median of the log-normal distribution.

The functions dLNO, pLNO, qLNO and rLNO define the density, distribution function, quantile function and random generation for the specific parameterization of the log-normal distribution and more generally a Box-Cox transformation.

Usage

LNO(mu.link = "identity", sigma.link = "log")
LOGNO(mu.link = "identity", sigma.link = "log")
LOGNO2(mu.link = "log", sigma.link = "log")
dLNO(x, mu = 1, sigma = 0.1, nu = 0, log = FALSE)
dLOGNO(x, mu = 0, sigma = 1, log = FALSE)
dLOGNO2(x, mu = 1, sigma = 1, log = FALSE)
pLNO(q, mu = 1, sigma = 0.1, nu = 0, lower.tail = TRUE, log.p = FALSE)
pLOGNO(q, mu = 0, sigma = 1, lower.tail = TRUE, log.p = FALSE)
pLOGNO2(q, mu = 1, sigma = 1, lower.tail = TRUE, log.p = FALSE)
qLNO(p, mu = 1, sigma = 0.1, nu = 0, lower.tail = TRUE, log.p = FALSE)
qLOGNO(p, mu = 0, sigma = 1, lower.tail = TRUE, log.p = FALSE)
qLOGNO2(p, mu = 1, sigma = 1, lower.tail = TRUE, log.p = FALSE)
rLNO(n, mu = 1, sigma = 0.1, nu = 0)
rLOGNO(n, mu = 0, sigma = 1)
rLOGNO2(n, mu = 1, sigma = 1)

Arguments

Details

The probability density function in LOGNO is defined as

f(y|\mu,\sigma)=\frac{1}{y \sqrt{2\pi}\sigma} \exp [-\frac{1}{2 \sigma^2}(\log y-\mu)^2 ]

for y>0, -\infty<\mu<\infty and \sigma>0 see pp. 428-429 of Rigby et al. (2019).

The probability density function in LOGNO2 is defined as

f(y|\mu,\sigma)=\frac{1}{y \sqrt{2\pi}\sigma} \exp [-\frac{1}{2 \sigma^2}(\log y-\log\mu)^2 ]

for y>0, -\infty<\mu<\infty and \sigma>0 see pp. 429-430 of Rigby et al. (2019).

The probability density function in LNO is defined as

f(y|\mu,\sigma,\nu)=\frac{y^{\nu-1}}{\sqrt{2\pi}\sigma} \exp [-\frac{1}{2 \sigma^2}(z-\mu)^2 ]

where if \nu \neq 0 z =(y^{\nu}-1)/\nu else z=\log(y) and z \sim N(0,\sigma^2), for y>0, \mu>0, \sigma>0 and \nu=(-\infty,+\infty). This is not a proper distribution see for example p. 447 of Rigby et al. (2019).

Value

LNO() returns a gamlss.family object which can be used to fit a log-normal distribution in the gamlss() function. dLNO() gives the density, pLNO() gives the distribution function, qLNO() gives the quantile function, and rLNO() generates random deviates.

Warning

This is a two parameter fit for \mu and \sigma while \nu is fixed. If you wish to model \nu use the gamlss family BCCG.

Note

\mu is the mean of z (and also the median of y), the Box-Cox transformed variable and \sigma is the standard deviation of z and approximate the coefficient of variation of y

Author(s)

Mikis Stasinopoulos, Bob Rigby and Calliope Akantziliotou

References

Box, G. E. P. and Cox, D. R. (1964) An analysis of transformations (with discussion), J. R. Statist. Soc. B., 26, 211–252

Rigby, R. A. and Stasinopoulos D. M. (2005). Generalized additive models for location, scale and shape,(with discussion), Appl. Statist., 54, part 3, pp 507-554.

Rigby, R. A., Stasinopoulos, D. M., Heller, G. Z., and De Bastiani, F. (2019) Distributions for modeling location, scale, and shape: Using GAMLSS in R, Chapman and Hall/CRC, doi:10.1201/9780429298547.An older version can be found in https://www.gamlss.com/.

Stasinopoulos D. M. Rigby R.A. (2007) Generalized additive models for location scale and shape (GAMLSS) in R. Journal of Statistical Software, Vol. 23, Issue 7, Dec 2007, doi:10.18637/jss.v023.i07.

Stasinopoulos D. M., Rigby R.A., Heller G., Voudouris V., and De Bastiani F., (2017) Stasinopoulos D. M., Rigby R.A., Heller G., Voudouris V., and De Bastiani F., (2017) Flexible Regression and Smoothing: Using GAMLSS in R, Chapman and Hall/CRC. doi:10.1201/b21973

(see also https://www.gamlss.com/).

See Also

gamlss.family, BCCG

Examples

LOGNO()#  gives information about the default links for the log normal distribution 
LOGNO2()
LNO()# gives information about the default links for the Box Cox distribution 

# plotting the d, p, q, and r functions
op<-par(mfrow=c(2,2))
curve(dLOGNO(x, mu=0), 0, 10)
curve(pLOGNO(x, mu=0), 0, 10)
curve(qLOGNO(x, mu=0), 0, 1)
Y<- rLOGNO(200)
hist(Y)
par(op)

# plotting the d, p, q, and r functions
op<-par(mfrow=c(2,2))
curve(dLOGNO2(x, mu=1), 0, 10)
curve(pLOGNO2(x, mu=1), 0, 10)
curve(qLOGNO2(x, mu=1), 0, 1)
Y<- rLOGNO(200)
hist(Y)
par(op)

# library(gamlss)
# data(abdom)
# h1<-gamlss(y~cs(x), family=LOGNO, data=abdom)#fits the log-Normal distribution  
# h2<-gamlss(y~cs(x), family=LNO, data=abdom)  #should be identical to the one above   
# to change to square root transformation, i.e. fix nu=0.5 
# h3<-gamlss(y~cs(x), family=LNO, data=abdom, nu.fix=TRUE, nu.start=0.5)

Logistic distribution for fitting a GAMLSS

Description

The function LO(), or equivalently Logistic(), defines the logistic distribution, a two parameter distribution, for a gamlss.family object to be used in GAMLSS fitting using the function gamlss()

Usage

LO(mu.link = "identity", sigma.link = "log")
dLO(x, mu = 0, sigma = 1, log = FALSE)
pLO(q, mu = 0, sigma = 1, lower.tail = TRUE, log.p = FALSE)
qLO(p, mu = 0, sigma = 1, lower.tail = TRUE, log.p = FALSE)
rLO(n, mu = 0, sigma = 1)

Arguments

Details

Definition file for Logistic distribution.

f(y|\mu,\sigma)=\frac{1}{\sigma} e^{-\left(\frac{y-\mu}{\sigma}\right)} [1+e^{-\left(\frac{y-\mu}{\sigma}\right)}]^{-2}

for y=(-\infty,\infty), \mu=(-\infty,\infty) and \sigma>0, see page 368 of Rigby et al. (2019).

Value

LO() returns a gamlss.family object which can be used to fit a logistic distribution in the gamlss() function. dLO() gives the density, pLO() gives the distribution function, qLO() gives the quantile function, and rLO() generates random deviates for the logistic distribution. The latest functions are based on the equivalent R functions for logistic distribution.

Note

\mu is the mean and \sigma \pi/ \sqrt3 is the standard deviation for the logistic distribution

Author(s)

Mikis Stasinopoulos, Bob Rigby and Calliope Akantziliotou

References

Rigby, R. A. and Stasinopoulos D. M. (2005). Generalized additive models for location, scale and shape,(with discussion), Appl. Statist., 54, part 3, pp 507-554.

Rigby, R. A., Stasinopoulos, D. M., Heller, G. Z., and De Bastiani, F. (2019) Distributions for modeling location, scale, and shape: Using GAMLSS in R, Chapman and Hall/CRC, doi:10.1201/9780429298547. An older version can be found in https://www.gamlss.com/.

Stasinopoulos D. M. Rigby R.A. (2007) Generalized additive models for location scale and shape (GAMLSS) in R. Journal of Statistical Software, Vol. 23, Issue 7, Dec 2007, doi:10.18637/jss.v023.i07.

Stasinopoulos D. M., Rigby R.A., Heller G., Voudouris V., and De Bastiani F., (2017) Flexible Regression and Smoothing: Using GAMLSS in R, Chapman and Hall/CRC. doi:10.1201/b21973

(see also https://www.gamlss.com/).

See Also

gamlss.family, NO, TF

Examples

LO()# gives information about the default links for the Logistic distribution 
plot(function(y) dLO(y, mu=10 ,sigma=2), 0, 20)
plot(function(y) pLO(y, mu=10 ,sigma=2), 0, 20)
plot(function(y) qLO(y, mu=10 ,sigma=2), 0, 1)
# library(gamlss)
# data(abdom)
# h<-gamlss(y~cs(x,df=3), sigma.formula=~cs(x,1), family=LO, data=abdom) # fits 
# plot(h)

Logit Normal distribution for fitting in GAMLSS

Description

The functions dLOGITNO, pLOGITNO, qLOGITNO and rLOGITNO define the density, distribution function, quantile function and random generation for the logit-normal distribution. The function LOGITNO can be used for fitting the distribution in gamlss().

Usage

LOGITNO(mu.link = "logit", sigma.link = "log")
dLOGITNO(x, mu = 0.5, sigma = 1, log = FALSE)
pLOGITNO(q, mu = 0.5, sigma = 1, lower.tail = TRUE, log.p = FALSE)
qLOGITNO(p, mu = 0.5, sigma = 1, lower.tail = TRUE, log.p = FALSE)
rLOGITNO(n, mu = 0.5, sigma = 1)

Arguments

Details

The probability density function in LOGITNO is defined as

f(y|\mu,\sigma)= \frac{1}{\sqrt{2\pi\sigma^2} y (1-y)} \exp \left( -\frac{1}{2 \sigma^2} \left[ \log(y/(1-y)-\log(\mu/(1-\mu)\right]^2 \right)

for 0<y<1, 0<\mu< 1 and \sigma>0 see p 463 of Rigby et al. (2019).

Value

LOGITNO() returns a gamlss.family object which can be used to fit a logit-normal distribution in the gamlss() function.

Author(s)

Mikis Stasinopoulos, Bob Rigby

References

Rigby, R. A. and Stasinopoulos D. M. (2005). Generalized additive models for location, scale and shape,(with discussion), Appl. Statist., 54, part 3, pp 507-554.

Rigby, R. A., Stasinopoulos, D. M., Heller, G. Z., and De Bastiani, F. (2019) Distributions for modeling location, scale, and shape: Using GAMLSS in R, Chapman and Hall/CRC, doi:10.1201/9780429298547. An older version can be found in https://www.gamlss.com/.

Stasinopoulos D. M. Rigby R.A. (2007) Generalized additive models for location scale and shape (GAMLSS) in R. Journal of Statistical Software, Vol. 23, Issue 7, Dec 2007, \doi10.18637/jss.v023.i07.

Stasinopoulos D. M., Rigby R.A., Heller G., Voudouris V., and De Bastiani F., (2017) Flexible Regression and Smoothing: Using GAMLSS in R, Chapman and Hall/CRC. doi:10.1201/b21973

(see also https://www.gamlss.com/).

See Also

gamlss.family, LOGNO

Examples

# plotting the d, p, q, and r functions
op<-par(mfrow=c(2,2))
curve(dLOGITNO(x), 0, 1)
curve(pLOGITNO(x), 0, 1)
curve(qLOGITNO(x), 0, 1)
Y<- rLOGITNO(200)
hist(Y)
par(op)

# plotting the d, p, q, and r functions
# sigma 3
op<-par(mfrow=c(2,2))
curve(dLOGITNO(x, sigma=3), 0, 1)
curve(pLOGITNO(x, sigma=3), 0, 1)
curve(qLOGITNO(x, sigma=3), 0, 1)
Y<- rLOGITNO(200, sigma=3)
hist(Y)
par(op)

Normal distribution with a specific mean and variance relationship for fitting a GAMLSS model

Description

The function LQNO() defines a normal distribution family, which has a specific mean and variance relationship. The distribution can be used in a GAMLSS fitting using the function gamlss(). The mean of LQNO is equal to mu. The variance is equal to mu*(1+sigma*mu) so the standard deviation is sqrt(mu*(1+sigma*mu)). The function is found useful in modelling small RNA sequencing experiments. The functions dLQNO, pLQNO, qLQNO and rLQNO define the density, distribution function, quantile function (inverse cdf) and random generation for the LQNO() parametrization of the normal distribution.

Usage

LQNO(mu.link = "log", sigma.link = "log")
dLQNO(x, mu = 1, sigma = 1, log = FALSE)
pLQNO(q, mu = 1, sigma = 1, lower.tail = TRUE, log.p = FALSE)
qLQNO(p, mu = 1, sigma = 1, lower.tail = TRUE, log.p = FALSE)
rLQNO(n, mu = 1, sigma = 1)

Arguments

Details

LQNO stands for Linear Quadratic Normal Family, in which the variance is a linear quadratic function of the mean: Var(Y) = mu*(1+sigma*mu). This is created to facilitate the analysis of data coming from small RNA sequencing experiments, basically counts of short RNAs that one isolates from cells or biofluids such as urine, plasma or cerebrospinal fluid. Argyropoulos et al. (2017) showing that the LQNO distribution (and the Negative Binomial which implements the same mean- variance relationship) are highly accurate approximations to the generative models of the signals in these experiments

Value

The function LQNO returns a gamlss.family object which can be used to fit this specific form of the normal distribution family in the gamlss() function.

Note

The mu parameters must be positive so for the relationship Var(Y) = mu*(1+sigma*mu) to be valid.

Author(s)

Christos Argyropoulos

References

Rigby, R. A., Stasinopoulos, D. M., Heller, G. Z., and De Bastiani, F. (2019) Distributions for modeling location, scale, and shape: Using GAMLSS in R, Chapman and Hall/CRC, doi:10.1201/9780429298547. An older version can be found in https://www.gamlss.com/.

Argyropoulos C, Etheridge A, Sakhanenko N, Galas D. (2017) Modeling bias and variation in the stochastic processes of small RNA sequencing. Nucleic Acids Res. 2017 Mar 27. doi: 10.1093/nar/gkx199. [Epub ahead of print] PubMed PMID: 28369495.

See Also

NO,NO2, NOF

Examples

LQNO()# gives information about the default links for the normal distribution
# a comparison of different Normal models
#m1 <- gamlss(y~pb(x), sigma.fo=~pb(x), data=abdom,  family=NO(mu.link="log"))
#m2 <- gamlss(y~pb(x), sigma.fo=~pb(x), data=abdom, family=LQNO)
#m3 <- gamlss(y~pb(x), sigma.fo=~pb(x), data=abdom, family=NOF(mu.link="log"))
#AIC(m1,m2,m3)

Multinomial distribution in GAMLSS

Description

The set of function presented here is useful for fitting multinomial regression within gamlss.

Usage

MN3(mu.link = "log", sigma.link = "log")
MN4(mu.link = "log", sigma.link = "log", nu.link = "log")
MN5(mu.link = "log", sigma.link = "log", nu.link = "log", tau.link = "log")
MULTIN(type = "3")
fittedMN(model)

dMN3(x, mu = 1, sigma = 1, log = FALSE)
dMN4(x, mu = 1, sigma = 1, nu = 1, log = FALSE)
dMN5(x, mu = 1, sigma = 1, nu = 1, tau = 1, log = FALSE)

pMN3(q, mu = 1, sigma = 1, lower.tail = TRUE, log.p = FALSE)
pMN4(q, mu = 1, sigma = 1, nu = 1, lower.tail = TRUE, log.p = FALSE)
pMN5(q, mu = 1, sigma = 1, nu = 1, tau = 1, lower.tail = TRUE, log.p = FALSE)

qMN3(p, mu = 1, sigma = 1, lower.tail = TRUE, log.p = FALSE)
qMN4(p, mu = 1, sigma = 1, nu = 1, lower.tail = TRUE, log.p = FALSE)
qMN5(p, mu = 1, sigma = 1, nu = 1, tau = 1, lower.tail = TRUE, log.p = FALSE)

rMN3(n, mu = 1, sigma = 1)
rMN4(n, mu = 1, sigma = 1, nu = 1)
rMN5(n, mu = 1, sigma = 1, nu = 1, tau = 1)

Arguments

Details

GAMLSS is in general not suitable for multinomial regression. Nevertheless multinomial regression can be fitted within GAMLSS if the response variable y has less than five categories. The function here provide the facilities to do so. The functions MN3(), MN4() and MN5() fit multinomial responses with 3, 4 and 5 categories respectively. The function MULTIN() can be used instead of codeMN3(), MN4() and MN5() by specifying the number of levels of the response. Note that MULTIN(2) will produce a binomial fit.

Value

returns a gamlss.family object which can be used to fit a binomial distribution in the gamlss() function.

Author(s)

Mikis Stasinopoulos, Bob Rigby and Vlasios Voudouris

References

Rigby, R. A. and Stasinopoulos D. M. (2005). Generalized additive models for location, scale and shape,(with discussion), Appl. Statist., 54, part 3, pp 507-554.

Rigby, R. A., Stasinopoulos, D. M., Heller, G. Z., and De Bastiani, F. (2019) Distributions for modeling location, scale, and shape: Using GAMLSS in R, Chapman and Hall/CRC, doi:10.1201/9780429298547. An older version can be found in https://www.gamlss.com/.

Stasinopoulos D. M. Rigby R.A. (2007) Generalized additive models for location scale and shape (GAMLSS) in R. Journal of Statistical Software, Vol. 23, Issue 7, Dec 2007, doi:10.18637/jss.v023.i07.

Stasinopoulos D. M., Rigby R.A., Heller G., Voudouris V., and De Bastiani F., (2017) Flexible Regression and Smoothing: Using GAMLSS in R, Chapman and Hall/CRC. doi:10.1201/b21973

(see also https://www.gamlss.com/).

See Also

gamlss.family, BI

Examples

 dMN3(3)
 pMN3(2)
 qMN3(.6)
 rMN3(10)
  

Negative Binomial Family distribution for fitting a GAMLSS

Description

The NBF() function defines the Negative Binomial family distribution, a three parameter distribution, for a gamlss.family object to be used in GAMLSS fitting using the function gamlss(). The functions dNBF, pNBF, qNBF and rNBF define the density, distribution function, quantile function and random generation for the negative binomial family, NBF(), distribution.

The functions dZINBF, pZINBF, qZINBF and rZINBF define the density, distribution function, quantile function and random generation for the zero inflated negative binomial family, ZINBF(), distribution a four parameter distribution.

Usage

NBF(mu.link = "log", sigma.link = "log", nu.link = "log")

dNBF(x, mu = 1, sigma = 1, nu = 2, log = FALSE)

pNBF(q, mu = 1, sigma = 1, nu = 2, lower.tail = TRUE, log.p = FALSE)

qNBF(p, mu = 1, sigma = 1, nu = 2, lower.tail = TRUE, log.p = FALSE)

rNBF(n, mu = 1, sigma = 1, nu = 2)

ZINBF(mu.link = "log", sigma.link = "log", nu.link = "log", 
      tau.link = "logit")
      
dZINBF(x, mu = 1, sigma = 1, nu = 2, tau = 0.1, log = FALSE)

pZINBF(q, mu = 1, sigma = 1, nu = 2, tau = 0.1, lower.tail = TRUE, 
      log.p = FALSE)
      
qZINBF(p, mu = 1, sigma = 1, nu = 2, tau = 0.1, lower.tail = TRUE, 
      log.p = FALSE)      
      
rZINBF(n, mu = 1, sigma = 1, nu = 2, tau = 0.1)      

Arguments

Details

The definition for Negative Binomial Family distribution , NBF, is similar to the Negative Binomial type I. The probability function of the NBF can be obtained by replacing \sigma with \sigma \mu^{\nu-2} where \nu is a power parameter. The distribution has mean \mu and variance \mu+\sigma \mu^{\nu}. For more details see pp 507-508 of Rigby et al. (2019).

The zero inflated negative binomial family ZINBF is defined as an inflated at zero NBF.

Value

returns a gamlss.family object which can be used to fit a Negative Binomial Family distribution in the gamlss() function.

Author(s)

Bob Rigby and Mikis Stasinopoulos

References

Anscombe, F. J. (1950) Sampling theory of the negative binomial and logarithmic distributions, Biometrika, 37, 358-382.

Rigby, R. A. and Stasinopoulos D. M. (2005). Generalized additive models for location, scale and shape,(with discussion), Appl. Statist., 54, part 3, pp 507-554.

Rigby, R. A., Stasinopoulos, D. M., Heller, G. Z., and De Bastiani, F. (2019) Distributions for modeling location, scale, and shape: Using GAMLSS in R, Chapman and Hall/CRC, doi:10.1201/9780429298547. An older version can be found in https://www.gamlss.com/.

Stasinopoulos D. M., Rigby R.A., Heller G., Voudouris V., and De Bastiani F., (2017) Flexible Regression and Smoothing: Using GAMLSS in R, Chapman and Hall/CRC. doi:10.1201/b21973

(see also https://www.gamlss.com/).

See Also

NBI, NBII

Examples

NBF() # default link functions for the Negative Binomial Family 
# plotting the distribution
plot(function(y) dNBF(y, mu = 10, sigma = 0.5, nu=2 ), from=0, 
     to=40, n=40+1, type="h")
# creating random variables and plot them 
tN <- table(Ni <- rNBF(1000, mu=5, sigma=0.5, nu=2))
r <- barplot(tN, col='lightblue')
# zero inflated NBF
ZINBF() # default link functions  for the zero inflated NBF 
# plotting the distribution
plot(function(y) dZINBF(y, mu = 10, sigma = 0.5, nu=2, tau=.1 ), 
     from=0, to=40, n=40+1, type="h")
# creating random variables and plot them 
tN <- table(Ni <- rZINBF(1000, mu=5, sigma=0.5, nu=2, tau=0.1))
r <- barplot(tN, col='lightblue')
## Not run: 
library(gamlss)
data(species)
species <- transform(species, x=log(lake))
m6 <- gamlss(fish~poly(x,2), sigma.fo=~1, data=species, family=NBF, 
          n.cyc=200)
fitted(m6, "nu")[1]

## End(Not run)

Negative Binomial type I distribution for fitting a GAMLSS

Description

The NBI() function defines the Negative Binomial type I distribution, a two parameter distribution, for a gamlss.family object to be used in GAMLSS fitting using the function gamlss(). The functions dNBI, pNBI, qNBI and rNBI define the density, distribution function, quantile function and random generation for the Negative Binomial type I, NBI(), distribution.

Usage

NBI(mu.link = "log", sigma.link = "log")
dNBI(x, mu = 1, sigma = 1, log = FALSE)
pNBI(q, mu = 1, sigma = 1, lower.tail = TRUE, log.p = FALSE)
qNBI(p, mu = 1, sigma = 1, lower.tail = TRUE, log.p = FALSE)
rNBI(n, mu = 1, sigma = 1)

Arguments

Details

Definition file for Negative Binomial type I distribution.

P(Y=y|\mu, \sigma)= \frac{\Gamma(y+\frac{1}{\sigma})}{\Gamma(\frac{1}{\sigma}) \Gamma(y+1)}\hspace{1mm}\left( \frac{\sigma \mu}{1+\sigma \mu}\right)^y \hspace{1mm}\left( \frac{1}{1+\sigma \mu} \right)^{1/\sigma}

for y=0,1,2,\ldots,\infty, \mu>0 and \sigma>0. This parameterization is equivalent to that used by Anscombe (1950) except he used \alpha=1/\sigma instead of \sigma, see also pp. 483-485 of Rigby et al. (2019).

Value

returns a gamlss.family object which can be used to fit a Negative Binomial type I distribution in the gamlss() function.

Warning

For values of \sigma<0.0001 the d,p,q,r functions switch to the Poisson distribution

Note

\mu is the mean and (\mu+\sigma \mu^2)^{0.5} is the standard deviation of the Negative Binomial type I distribution (so \sigma is the dispersion parameter in the usual GLM for the negative binomial type I distribution)

Author(s)

Mikis Stasinopoulos, Bob Rigby and Calliope Akantziliotou

References

Anscombe, F. J. (1950) Sampling theory of the negative bimomial and logarithmic distributiona, Biometrika, 37, 358-382.

Rigby, R. A. and Stasinopoulos D. M. (2005). Generalized additive models for location, scale and shape,(with discussion), Appl. Statist., 54, part 3, pp 507-554.

Rigby, R. A., Stasinopoulos, D. M., Heller, G. Z., and De Bastiani, F. (2019) Distributions for modeling location, scale, and shape: Using GAMLSS in R, Chapman and Hall/CRC, doi:10.1201/9780429298547. An older version can be found in https://www.gamlss.com/.

Stasinopoulos D. M. Rigby R.A. (2007) Generalized additive models for location scale and shape (GAMLSS) in R. Journal of Statistical Software, Vol. 23, Issue 7, Dec 2007, doi:10.18637/jss.v023.i07.

Stasinopoulos D. M., Rigby R.A., Heller G., Voudouris V., and De Bastiani F., (2017) Flexible Regression and Smoothing: Using GAMLSS in R, Chapman and Hall/CRC. doi:10.1201/b21973

(see also https://www.gamlss.com/).

See Also

gamlss.family, NBII, PIG, SI

Examples

NBI()   # gives information about the default links for the Negative Binomial type I distribution  
# plotting the distribution
plot(function(y) dNBI(y, mu = 10, sigma = 0.5 ), from=0, to=40, n=40+1, type="h")
# creating random variables and plot them 
tN <- table(Ni <- rNBI(1000, mu=5, sigma=0.5))
r <- barplot(tN, col='lightblue')
# library(gamlss)
# data(aids)
# h<-gamlss(y~cs(x,df=7)+qrt, family=NBI, data=aids) # fits the model 
# plot(h)
# pdf.plot(family=NBI, mu=10, sigma=0.5, min=0, max=40, step=1)

Negative Binomial type II distribution for fitting a GAMLSS

Description

The NBII() function defines the Negative Binomial type II distribution, a two parameter distribution, for a gamlss.family object to be used in GAMLSS fitting using the function gamlss(). The functions dNBII, pNBII, qNBII and rNBII define the density, distribution function, quantile function and random generation for the Negative Binomial type II, NBII(), distribution.

Usage

NBII(mu.link = "log", sigma.link = "log")
dNBII(x, mu = 1, sigma = 1, log = FALSE)
pNBII(q, mu = 1, sigma = 1, lower.tail = TRUE, log.p = FALSE)
qNBII(p, mu = 1, sigma = 1, lower.tail = TRUE, log.p = FALSE)
rNBII(n, mu = 1, sigma = 1)

Arguments

Details

Definition file for Negative Binomial type II distribution.

P(Y=y|\mu,\sigma)= \frac{\Gamma(y+\frac{\mu}{\sigma}) \sigma^y }{\Gamma(\frac{\mu}{\sigma})\Gamma(y+1) (1+\sigma)^{y+\mu/\sigma}}

for y=0,1,2,...,\infty, \mu>0 and \sigma>0. This parameterization was used by Evans (1953) and also by Johnson et al. (1993) p 200, see also pp. 485-487 of Rigby et al. (2019).

Value

returns a gamlss.family object which can be used to fit a Negative Binomial type II distribution in the gamlss() function.

Note

\mu is the mean and [(1+\sigma)\mu]^{0.5} is the standard deviation of the Negative Binomial type II distribution, so \sigma is a dispersion parameter

Author(s)

Mikis Stasinopoulos, Bob Rigby and Calliope Akantziliotou

References

Evans, D. A. (1953). Experimental evidence concerning contagious distributions in ecology. Biometrika, 40: 186-211.

Johnson, N. L., Kotz, S. and Kemp, A. W. (1993). Univariate Discrete Distributions, 2nd edn. Wiley, New York.

Rigby, R. A. and Stasinopoulos D. M. (2005). Generalized additive models for location, scale and shape,(with discussion), Appl. Statist., 54, part 3, pp 507-554.

Rigby, R. A., Stasinopoulos, D. M., Heller, G. Z., and De Bastiani, F. (2019) Distributions for modeling location, scale, and shape: Using GAMLSS in R, Chapman and Hall/CRC, doi:10.1201/9780429298547. An older version can be found in https://www.gamlss.com/.

Stasinopoulos D. M. Rigby R.A. (2007) Generalized additive models for location scale and shape (GAMLSS) in R. Journal of Statistical Software, Vol. 23, Issue 7, Dec 2007, doi:10.18637/jss.v023.i07.

Stasinopoulos D. M., Rigby R.A., Heller G., Voudouris V., and De Bastiani F., (2017) Flexible Regression and Smoothing: Using GAMLSS in R, Chapman and Hall/CRC. doi:10.1201/b21973

(see also https://www.gamlss.com/).

See Also

gamlss.family, NBI, PIG, SI

Examples

NBII()  # gives information about the default links for the Negative Binomial type II distribution  
# plotting the distribution
plot(function(y) dNBII(y, mu = 10, sigma = 0.5 ), from=0, to=40, n=40+1, type="h")
# creating random variables and plot them 
tN <- table(Ni <- rNBII(1000, mu=5, sigma=0.5))
r <- barplot(tN, col='lightblue')
# library(gamlss)
# data(aids)
# h<-gamlss(y~cs(x,df=7)+qrt, family=NBII, data=aids) # fits a model 
# plot(h)
# pdf.plot(family=NBII, mu=10, sigma=0.5, min=0, max=40, step=1)

Normal Exponential t distribution (NET) for fitting a GAMLSS

Description

This function defines the Power Exponential t distribution (NET), a four parameter distribution, for a gamlss.family object to be used for a GAMLSS fitting using the function gamlss(). The functions dNET, pNET define the density and distribution function the NET distribution.

Usage

NET(mu.link = "identity", sigma.link = "log", nu.link ="identity",
      tau.link = "identity")
pNET(q, mu=0, sigma=1, nu=1.5, tau=2,  lower.tail = TRUE, log.p = FALSE)
dNET(x, mu=0, sigma=1, nu=1.5, tau=2, log=FALSE)
qNET(p, mu=0, sigma=1, nu=1.5, tau=2,  lower.tail = TRUE, log.p = FALSE)
rNET(n,  mu=0, sigma=1, nu=1.5, tau=2)

Arguments

Details

The NET distribution was introduced by Rigby and Stasinopoulos (1994) as a robust distribution for a response variable with heavier tails than the normal. The NET distribution is the abbreviation of the Normal Exponential Student t distribution. The NET distribution is a four parameter continuous distribution, although in the GAMLSS implementation only the two parameters, mu and sigma, of the distribution are modelled with nu and tau fixed. The distribution takes its names because it is normal up to nu, Exponential from nu to tau (hence abs(nu)<=abs(tau)) and Student-t with nu*tau-1 degrees of freedom after tau. Maximum likelihood estimator of the third and forth parameter can be obtained, using the GAMLSS functions, find.hyper or prof.dev.

For more details about the NET distribution please refer to pp. 393-396 of of Rigby et al. (2019).

Value

NET() returns a gamlss.family object which can be used to fit a Box Cox Power Exponential distribution in the gamlss() function. dNET() gives the density, pNET() gives the distribution function.

Author(s)

Mikis Stasinopoulos, Bob Rigby and Calliope Akantziliotou

References

Rigby, R. A. and Stasinopoulos, D. M. (1994), Robust fitting of an additive model for variance heterogeneity, COMPSTAT : Proceedings in Computational Statistics, editors:R. Dutter and W. Grossmann, pp 263-268, Physica, Heidelberg.

Rigby, R. A. and Stasinopoulos D. M. (2005). Generalized additive models for location, scale and shape,(with discussion), Appl. Statist., 54, part 3, pp 507-554.

Rigby, R. A., Stasinopoulos, D. M., Heller, G. Z., and De Bastiani, F. (2019) Distributions for modeling location, scale, and shape: Using GAMLSS in R, Chapman and Hall/CRC, doi:10.1201/9780429298547. An older version can be found in https://www.gamlss.com/.

Stasinopoulos D. M. Rigby R.A. (2007) Generalized additive models for location scale and shape (GAMLSS) in R. Journal of Statistical Software, Vol. 23, Issue 7, Dec 2007, doi:10.18637/jss.v023.i07.

SStasinopoulos D. M., Rigby R.A., Heller G., Voudouris V., and De Bastiani F., (2017) Flexible Regression and Smoothing: Using GAMLSS in R, Chapman and Hall/CRC. doi:10.1201/b21973

(see also https://www.gamlss.com/).

See Also

gamlss.family, BCPE

Examples

NET()   # 
data(abdom)
plot(function(x)dNET(x, mu=0,sigma=1,nu=2, tau=3), -5, 5)
plot(function(x)pNET(x, mu=0,sigma=1,nu=2, tau=3), -5, 5) 
# fit NET with nu=1 and tau=3
# library(gamlss)
#h<-gamlss(y~cs(x,df=3), sigma.formula=~cs(x,1), family=NET, 
#        data=abdom, nu.start=2, tau.start=3) 
#plot(h)

Normal distribution for fitting a GAMLSS

Description

The function NO() defines the normal distribution, a two parameter distribution, for a gamlss.family object to be used in GAMLSS fitting using the function gamlss(), with mean equal to the parameter mu and sigma equal the standard deviation. The functions dNO, pNO, qNO and rNO define the density, distribution function, quantile function and random generation for the NO parameterization of the normal distribution. [A alternative parameterization with sigma equal to the variance is given in the function NO2()]

Usage

NO(mu.link = "identity", sigma.link = "log")
dNO(x, mu = 0, sigma = 1, log = FALSE)
pNO(q, mu = 0, sigma = 1, lower.tail = TRUE, log.p = FALSE)
qNO(p, mu = 0, sigma = 1, lower.tail = TRUE, log.p = FALSE)
rNO(n, mu = 0, sigma = 1)

Arguments

Details

The parametrization of the normal distribution given in the function NO() is

f(y|\mu,\sigma)=\frac{1}{\sqrt{2 \pi }\sigma}\exp \left[-\frac{1}{2}(\frac{y-\mu}{\sigma})^2\right]

for y=(-\infty,\infty), \mu=(-\infty,+\infty) and \sigma>0 see pp. 369-370 of Rigby et al. (2019).

Value

returns a gamlss.family object which can be used to fit a normal distribution in the gamlss() function.

Note

For the function NO(), \mu is the mean and \sigma is the standard deviation (not the variance) of the normal distribution.

Author(s)

Mikis Stasinopoulos, Bob Rigby and Calliope Akantziliotou

References

Rigby, R. A. and Stasinopoulos D. M. (2005). Generalized additive models for location, scale and shape,(with discussion), Appl. Statist., 54, part 3, pp 507-554.

Rigby, R. A., Stasinopoulos, D. M., Heller, G. Z., and De Bastiani, F. (2019) Distributions for modeling location, scale, and shape: Using GAMLSS in R, Chapman and Hall/CRC, doi:10.1201/9780429298547. An older version can be found in https://www.gamlss.com/.

Stasinopoulos D. M. Rigby R.A. (2007) Generalized additive models for location scale and shape (GAMLSS) in R. Journal of Statistical Software, Vol. 23, Issue 7, Dec 2007, doi:10.18637/jss.v023.i07.

Stasinopoulos D. M., Rigby R.A., Heller G., Voudouris V., and De Bastiani F., (2017) Flexible Regression and Smoothing: Using GAMLSS in R, Chapman and Hall/CRC. doi:10.1201/b21973

(see also https://www.gamlss.com/).

See Also

gamlss.family, NO2

Examples

NO()# gives information about the default links for the normal distribution
plot(function(y) dNO(y, mu=10 ,sigma=2), 0, 20)
plot(function(y) pNO(y, mu=10 ,sigma=2), 0, 20)
plot(function(y) qNO(y, mu=10 ,sigma=2), 0, 1)
dat<-rNO(100)
hist(dat)
# library(gamlss)        
# gamlss(dat~1,family=NO) # fits a constant for mu and sigma 

Normal distribution (with variance as sigma parameter) for fitting a GAMLSS

Description

The function NO2() defines the normal distribution, a two parameter distribution, for a gamlss.family object to be used in GAMLSS fitting using the function gamlss() with mean equal to mu and variance equal to sigma. The functions dNO2, pNO2, qNO2 and rNO2 define the density, distribution function, quantile function and random generation for this specific parameterization of the normal distribution.

[A alternative parameterization with sigma as the standard deviation is given in the function NO()]

Usage

NO2(mu.link = "identity", sigma.link = "log")
dNO2(x, mu = 0, sigma = 1, log = FALSE)
pNO2(q, mu = 0, sigma = 1, lower.tail = TRUE, log.p = FALSE)
qNO2(p, mu = 0, sigma = 1, lower.tail = TRUE, log.p = FALSE)
rNO2(n, mu = 0, sigma = 1)

Arguments

Details

The parametrization of the normal distribution given in the function NO2() is

f(y|\mu,\sigma)=\frac{1}{\sqrt{2 \pi \sigma}}\exp\left[-\frac{1}{2}\frac{(y-\mu)^2}{\sigma}\right]

for y=(-\infty,\infty), \mu=(-\infty,+\infty) and \sigma>0 see p. 370 of Rigby et al. (2019).

Value

returns a gamlss.family object which can be used to fit a normal distribution in the gamlss() function.

Note

For the function NO(), \mu is the mean and \sigma is the standard deviation (not the variance) of the normal distribution. [The function NO2() defines the normal distribution with \sigma as the variance.]

Author(s)

Mikis Stasinopoulos, Bob Rigby and Calliope Akantziliotou

References

Rigby, R. A. and Stasinopoulos D. M. (2005). Generalized additive models for location, scale and shape,(with discussion), Appl. Statist., 54, part 3, pp 507-554.

Rigby, R. A., Stasinopoulos, D. M., Heller, G. Z., and De Bastiani, F. (2019) Distributions for modeling location, scale, and shape: Using GAMLSS in R, Chapman and Hall/CRC, doi:10.1201/9780429298547. An older version can be found in https://www.gamlss.com/.

Stasinopoulos D. M. Rigby R.A. (2007) Generalized additive models for location scale and shape (GAMLSS) in R. Journal of Statistical Software, Vol. 23, Issue 7, Dec 2007, doi:10.18637/jss.v023.i07.

Stasinopoulos D. M., Rigby R.A., Heller G., Voudouris V., and De Bastiani F., (2017) Flexible Regression and Smoothing: Using GAMLSS in R, Chapman and Hall/CRC. doi:10.1201/b21973

(see also https://www.gamlss.com/).

See Also

gamlss.family, NO

Examples

NO()# gives information about the default links for the normal distribution
dat<-rNO(100)
hist(dat)        
plot(function(y) dNO(y, mu=10 ,sigma=2), 0, 20)
plot(function(y) pNO(y, mu=10 ,sigma=2), 0, 20)
plot(function(y) qNO(y, mu=10 ,sigma=2), 0, 1)
# library(gamlss)
# gamlss(dat~1,family=NO) # fits a constant for mu and sigma 

Normal distribution family for fitting a GAMLSS

Description

The function NOF() defines a normal distribution family, which has three parameters. The distribution can be used using the function gamlss(). The mean of NOF is equal to mu. The variance is equal to sigma^2*mu^nu so the standard deviation is sigma*mu^(nu/2). The function is design for cases where the variance is proportional to a power of the mean. This is an instance of the Taylor's power low, see Enki et al. (2017). The functions dNOF, pNOF, qNOF and rNOF define the density, distribution function, quantile function and random generation for the NOF parametrization of the normal distribution family.

Usage

NOF(mu.link = "identity", sigma.link = "log", nu.link = "identity")
dNOF(x, mu = 0, sigma = 1, nu = 0, log = FALSE)
pNOF(q, mu = 0, sigma = 1, nu = 0, lower.tail = TRUE, log.p = FALSE)
qNOF(p, mu = 0, sigma = 1, nu = 0, lower.tail = TRUE, log.p = FALSE)
rNOF(n, mu = 0, sigma = 1, nu = 0)

Arguments

Details

The parametrization of the normal distribution given in the function NOF() is

f(y|\mu,\sigma, \nu)=\frac{1}{\sqrt{2 \pi }\sigma \mu^{\nu/2}}\exp \left[-\frac{1}{2}\frac{(y-\mu)^2}{\sigma^2 \mu^\nu}\right]

for y=(-\infty,\infty), \mu=(-\infty,\infty), \sigma>0 and \nu=(-\infty,+\infty) see pp. 373-374 of Rigby et al. (2019).

Value

returns a gamlss.family object which can be used to fit a normal distribution family in the gamlss() function.

Note

For the function NOF(), \mu is the mean and \sigma \mu^{\nu/2} is the standard deviation of the normal distribution family. The NOF is design for fitting regression type models where the variance is proportional to a power of the mean. Models of this type are also related to the "pseudo likelihood" models of Carroll and Rubert (1987) but here a proper likelihood is miximised.

Note that because the high correlation between the sigma and the nu parameter the mixed() method should be used in the fitting.

Author(s)

Mikis Stasinopoulos, Bob Rigby and Calliope Akantziliotou

References

Davidian, M. and Carroll, R. J. (1987), Variance Function Estimation, Journal of the American Statistical Association, Vol. 82, pp. 1079-1091

Enki, D G, Noufaily, A., Farrington, P., Garthwaite, P., Andrews, N. and Charlett, A. (2017) Taylor's power law and the statistical modelling of infectious disease surveillance data, Journal of the Royal Statistical Society: Series A (Statistics in Society), volume=180, number=1, pages=45-72.

Rigby, R. A. and Stasinopoulos D. M. (2005). Generalized additive models for location, scale and shape,(with discussion), Appl. Statist., 54, part 3, pp 507-554.

Rigby, R. A., Stasinopoulos, D. M., Heller, G. Z., and De Bastiani, F. (2019) Distributions for modeling location, scale, and shape: Using GAMLSS in R, Chapman and Hall/CRC, doi:10.1201/9780429298547. An older version can be found in https://www.gamlss.com/.

Stasinopoulos D. M. Rigby R.A. (2007) Generalized additive models for location scale and shape (GAMLSS) in R. Journal of Statistical Software, Vol. 23, Issue 7, Dec 2007, doi:10.18637/jss.v023.i07.

Stasinopoulos D. M., Rigby R.A., Heller G., Voudouris V., and De Bastiani F., (2017) Flexible Regression and Smoothing: Using GAMLSS in R, Chapman and Hall/CRC. doi:10.1201/b21973

(see also https://www.gamlss.com/).

See Also

gamlss.family, NO, NO2

Examples

NOF()# gives information about the default links for the normal distribution family
## Not run: 
## the normal distribution, fitting a constant sigma
m1<-gamlss(y~poly(x,2), sigma.fo=~1, family=NO, data=abdom)
## the normal family, fitting a variance proportional to the mean (mu)
m2<-gamlss(y~poly(x,2), sigma.fo=~1, family=NOF, data=abdom, method=mixed(1,20))
## the nornal distribution fitting  the variance as a function of x
m3 <-gamlss(y~poly(x,2), sigma.fo=~x,   family=NO, data=abdom, method=mixed(1,20)) 
GAIC(m1,m2,m3)

## End(Not run)

Pareto distributions for fitting in GAMLSS

Description

The functions PARETO() defines the one parameter Pareto distribution for y>1.

The functions PARETO1() defines the one parameter Pareto distribution for y>0.

The functions PARETOo1() defines the one parameter Pareto distribution for y>mu therefor requires mu to be fixed.

The functions PARETO2() and PARETO2o() define the Pareto Type 2 distribution, for y>0, a two parameter distribution, for a gamlss.family object to be used in GAMLSS fitting using the function gamlss(). The parameters are mu and sigma in both functions but the parameterasation different. The mu is identical for both PARETO2() and PARETO2o(). The sigma in PARETO2o() is the inverse of the sigma in codePARETO2() and coresponse to the usual parameter alpha of the Patreto distribution. The functions dPARETO2, pPARETO2, qPARETO2 and rPARETO2 define the density, distribution function, quantile function and random generation for the PARETO2 parameterization of the Pareto type 2 distribution while the functions dPARETO2o, pPARETO2o, qPARETO2o and rPARETO2o define the density, distribution function, quantile function and random generation for the original PARETO2o parameterization of the Pareto type 2 distribution

Usage

PARETO(mu.link = "log")
dPARETO(x, mu = 1, log = FALSE)
pPARETO(q, mu = 1, lower.tail = TRUE, log.p = FALSE)
qPARETO(p, mu = 1, lower.tail = TRUE, log.p = FALSE)
rPARETO(n, mu = 1)

PARETO1(mu.link = "log")
dPARETO1(x, mu = 1, log = FALSE)
pPARETO1(q, mu = 1, lower.tail = TRUE, log.p = FALSE)
qPARETO1(p, mu = 1, lower.tail = TRUE, log.p = FALSE)
rPARETO1(n, mu = 1)

PARETO1o(mu.link = "log", sigma.link = "log")
dPARETO1o(x, mu = 1, sigma = 0.5, log = FALSE)
pPARETO1o(q, mu = 1, sigma = 0.5, lower.tail = TRUE, log.p = FALSE)
qPARETO1o(p, mu = 1, sigma = 0.5, lower.tail = TRUE, log.p = FALSE)
rPARETO1o(n, mu = 1, sigma = 0.5)

PARETO2(mu.link = "log", sigma.link = "log")
dPARETO2(x, mu = 1, sigma = 0.5, log = FALSE)
pPARETO2(q, mu = 1, sigma = 0.5, lower.tail = TRUE, log.p = FALSE)
qPARETO2(p, mu = 1, sigma = 0.5, lower.tail = TRUE, log.p = FALSE)
rPARETO2(n, mu = 1, sigma = 0.5)

PARETO2o(mu.link = "log", sigma.link = "log")
dPARETO2o(x, mu = 1, sigma = 0.5, log = FALSE)
pPARETO2o(q, mu = 1, sigma = 0.5, lower.tail = TRUE, log.p = FALSE)
qPARETO2o(p, mu = 1, sigma = 0.5, lower.tail = TRUE, log.p = FALSE)
rPARETO2o(n, mu = 1, sigma = 0.5)

Arguments

Details

The parameterization of the one parameter Pareto distribution in the function PARETO is:

f(y|\mu) = \mu y^{\mu+1}

for y>1 and \mu>0.

The parameterization of the Pareto Type 1 original distribution in the function PARETO1o is:

f(y|\mu, \sigma) = \frac{\sigma \mu^{\sigma}}{y^{\sigma+1}}

for y>=0, \mu>0 and \sigma>0 see pp. 430-431 of Rigby et al. (2019).

The parameterization of the Pareto Type 2 original distribution in the function PARETO2o is:

f(y|\mu, \sigma) = \frac{\sigma \mu^{\sigma}}{(y+\mu)^{\sigma+1}}

for y>=0, \mu>0 and \sigma>0 see pp. 432-433 of Rigby et al. (2019).

The parameterization of the Pareto Type 2 distribution in the function PARETO2 is:

f(y|\mu, \sigma) = \frac{1}{\sigma} \mu^{\frac{1}{\sigma}} \, (y+\mu)^{-\frac{1 }{\sigma+1}}

for y>=0, \mu>0 and \sigma>0 see pp.433-434 The parameterization of the Pareto Type 1 original distribution in the function PARETO1o is:

f(y|\mu, \sigma) = \frac{\sigma \mu^{\sigma}}{y^{\sigma+1}}

for y>=0, \mu>0 and \sigma>0 see pp. 430-431 of Rigby et al. (2019).

Value

returns a gamlss.family object which can be used to fit a Pareto type 2 distribution in the gamlss() function.

Author(s)

Fiona McElduff, Bob Rigby and Mikis Stasinopoulos

References

Johnson, N., Kotz, S., and Balakrishnan, N. (1997). Discrete Multivariate Distributions. Wiley-Interscience, NY, USA.

Rigby, R. A. and Stasinopoulos D. M. (2005). Generalized additive models for location, scale and shape,(with discussion), Appl. Statist., 54, part 3, pp 507-554.

Rigby, R. A., Stasinopoulos, D. M., Heller, G. Z., and De Bastiani, F. (2019) Distributions for modeling location, scale, and shape: Using GAMLSS in R, Chapman and Hall/CRC, doi:10.1201/9780429298547. An older version can be found in https://www.gamlss.com/.

Stasinopoulos D. M. Rigby R.A. (2007) Generalized additive models for location scale and shape (GAMLSS) in R. Journal of Statistical Software, Vol. 23, Issue 7, Dec 2007, doi:10.18637/jss.v023.i07.

Stasinopoulos D. M., Rigby R.A., Heller G., Voudouris V., and De Bastiani F., (2017) Flexible Regression and Smoothing: Using GAMLSS in R, Chapman and Hall/CRC. doi:10.1201/b21973

(see also https://www.gamlss.com/).

See Also

gamlss.family

Examples

par(mfrow=c(2,2))
y<-seq(0.2,20,0.2)
plot(y, dPARETO2(y), type="l" , lwd=2)
q<-seq(0,20,0.2)
plot(q, pPARETO2(q), ylim=c(0,1), type="l", lwd=2) 
p<-seq(0.0001,0.999,0.05)
plot(p, qPARETO2(p), type="l", lwd=2)
dat <- rPARETO2(100)
hist(rPARETO2(100), nclass=30)
#summary(gamlss(a~1, family="PARETO2"))

Power Exponential distribution for fitting a GAMLSS

Description

The functions define the Power Exponential distribution, a three parameter distribution, for a gamlss.family object to be used in GAMLSS fitting using the function gamlss(). The functions dPE, pPE, qPE and rPE define the density, distribution function, quantile function and random generation for the specific parameterization of the power exponential distribution showing below. The functions dPE2, pPE2, qPE2 and rPE2 define the density, distribution function, quantile function and random generation of a standard parameterization of the power exponential distribution.

Usage

PE(mu.link = "identity", sigma.link = "log", nu.link = "log")
dPE(x, mu = 0, sigma = 1, nu = 2, log = FALSE)
pPE(q, mu = 0, sigma = 1, nu = 2, lower.tail = TRUE, log.p = FALSE)
qPE(p, mu = 0, sigma = 1, nu = 2, lower.tail = TRUE, log.p = FALSE)
rPE(n, mu = 0, sigma = 1, nu = 2)
PE2(mu.link = "identity", sigma.link = "log", nu.link = "log")
dPE2(x, mu = 0, sigma = 1, nu = 2, log = FALSE)
pPE2(q, mu = 0, sigma = 1, nu = 2, lower.tail = TRUE, log.p = FALSE)
qPE2(p, mu = 0, sigma = 1, nu = 2, lower.tail = TRUE, log.p = FALSE)
rPE2(n, mu = 0, sigma = 1, nu = 2)

Arguments

Details

Power Exponential distribution (PE) is defined as:

f(y|\mu,\sigma,\nu)=\frac{\nu \exp[- |z|^{\nu}]}{2 c \sigma \Gamma(\frac{1}{\nu})}

where z=(y-\mu)/ c \sigma and c^2=\Gamma(1/\nu)\left[/\Gamma(3/\nu) \right]^{-1}, for y=(-\infty,+\infty), \mu=(-\infty,+\infty), \sigma>0 and \nu>0. This parametrization was used by Nelson (1991) and ensures \mu is the mean and \sigma is the standard deviation of y (for all parameter values of \mu, \sigma and \nu within the ranges above), see p. 374 of Rigby et al. (2019)

Thw Power Exponential distribution (PE2) is defined as

f(y|\mu,\sigma,\nu)=\frac{\nu \exp[-\left|z\right|^\nu]} {2\sigma \Gamma\left(\frac{1}{\nu}\right)}

see p. 376 of Rigby et al. (2019)

Value

returns a gamlss.family object which can be used to fit a Power Exponential distribution in the gamlss() function.

Note

\mu is the mean and \sigma is the standard deviation of the Power Exponential distribution

Author(s)

Mikis Stasinopoulos, Bob Rigby

References

Nelson, D.B. (1991) Conditional heteroskedasticity in asset returns: a new approach. Econometrica, 57, 347-370.

Rigby, R. A. and Stasinopoulos D. M. (2005). Generalized additive models for location, scale and shape,(with discussion), Appl. Statist., 54, part 3, pp 507-554.

Rigby, R. A., Stasinopoulos, D. M., Heller, G. Z., and De Bastiani, F. (2019) Distributions for modeling location, scale, and shape: Using GAMLSS in R, Chapman and Hall/CRC, doi:10.1201/9780429298547. An older version can be found in https://www.gamlss.com/.

Stasinopoulos D. M. Rigby R.A. (2007) Generalized additive models for location scale and shape (GAMLSS) in R. Journal of Statistical Software, Vol. 23, Issue 7, Dec 2007, doi:10.18637/jss.v023.i07.

Stasinopoulos D. M., Rigby R.A., Heller G., Voudouris V., and De Bastiani F., (2017) Flexible Regression and Smoothing: Using GAMLSS in R, Chapman and Hall/CRC. doi:10.1201/b21973

(see also https://www.gamlss.com/).

See Also

gamlss.family, BCPE

Examples

PE()# gives information about the default links for the Power Exponential distribution  
# library(gamlss)
# data(abdom)
# h1<-gamlss(y~cs(x,df=3), sigma.formula=~cs(x,1), family=PE, data=abdom) # fit
# h2<-gamlss(y~cs(x,df=3), sigma.formula=~cs(x,1), family=PE2, data=abdom) # fit 
# plot(h1)
# plot(h2)
# leptokurtotic
plot(function(x) dPE(x, mu=10,sigma=2,nu=1), 0.0, 20, 
 main = "The PE  density mu=10,sigma=2,nu=1")
# platykurtotic
plot(function(x) dPE(x, mu=10,sigma=2,nu=4), 0.0, 20, 
 main = "The PE  density mu=10,sigma=2,nu=4") 

The Poisson-inverse Gaussian distribution for fitting a GAMLSS model

Description

The PIG() function defines the Poisson-inverse Gaussian distribution, a two parameter distribution, for a gamlss.family object to be used in GAMLSS fitting using the function gamlss(). The PIG2() function is a repametrization of PIG() where mu and sigma are orthogonal see Heller et al. (2018).

The functions dPIG, pPIG, qPIG and rPIG define the density, distribution function, quantile function and random generation for the Poisson-inverse Gaussian PIG(), distribution. Also codedPIG2, pPIG2, qPIG2 and rPIG2 are the equivalent functions for codePIG2()

The functions ZAPIG() and ZIPIG() are the zero adjusted (hurdle) and zero inflated versions of the Poisson-inverse Gaussian distribution, respectively. That is three parameter distributions.

The functions dZAPIG, dZIPIG, pZAPIG,pZIPIG, qZAPIG qZIPIG rZAPIG and rZIPIG define the probability, cumulative, quantile and random generation functions for the zero adjusted and zero inflated Poisson Inverse Gaussian distributions, ZAPIG(), ZIPIG(), respectively.

Usage

PIG(mu.link = "log", sigma.link = "log")
dPIG(x, mu = 1, sigma = 1, log = FALSE)
pPIG(q, mu = 1, sigma = 1, lower.tail = TRUE, log.p = FALSE)
qPIG(p, mu = 1, sigma = 1, lower.tail = TRUE, log.p = FALSE, 
     max.value = 10000)
rPIG(n, mu = 1, sigma = 1, max.value = 10000)

PIG2(mu.link = "log", sigma.link = "log")
dPIG2(x, mu=0.5, sigma=0.02, log = FALSE)
pPIG2(q, mu=0.5, sigma=0.02, lower.tail = TRUE, log.p = FALSE)
qPIG2(p, mu=0.5, sigma=0.02, lower.tail = TRUE, log.p = FALSE, 
     max.value = 10000)
rPIG2(n, mu=0.5, sigma=0.02)

ZIPIG(mu.link = "log", sigma.link = "log", nu.link = "logit")
dZIPIG(x, mu = 1, sigma = 1, nu = 0.3, log = FALSE)
pZIPIG(q, mu = 1, sigma = 1, nu = 0.3, lower.tail = TRUE, log.p = FALSE)
qZIPIG(p, mu = 1, sigma = 1, nu = 0.3, lower.tail = TRUE, log.p = FALSE, 
       max.value = 10000)
rZIPIG(n, mu = 1, sigma = 1, nu = 0.3, max.value = 10000)

ZAPIG(mu.link = "log", sigma.link = "log", nu.link = "logit")
dZAPIG(x, mu = 1, sigma = 1, nu = 0.3, log = FALSE)
pZAPIG(q, mu = 1, sigma = 1, nu = 0.3, lower.tail = TRUE, log.p = FALSE)
qZAPIG(p, mu = 1, sigma = 1, nu = 0.3, lower.tail = TRUE, log.p = FALSE, 
      max.value = 10000)
rZAPIG(n, mu = 1, sigma = 1, nu = 0.3, max.value = 10000)

Arguments

Details

The probability function of the Poisson-inverse Gaussian distribution PIG, is given by

f(y|\mu,\sigma)=\left( \frac{2 \alpha}{\pi}^{\frac{1}{2}}\right)\frac{\mu^y e^{\frac{1}{\sigma}} K_{y-\frac{1}{2}}(\alpha)}{(\alpha \sigma)^y y!}

where \alpha^2=\frac{1}{\sigma^2}+\frac{2\mu}{\sigma}, for y=0,1,2,...,\infty where \mu>0 and \sigma>0 and K_{\lambda}(t)=\frac{1}{2}\int_0^{\infty} x^{\lambda-1} \exp\{-\frac{1}{2}t(x+x^{-1})\}dx is the modified Bessel function of the third kind, see also pp. 487-489 of Rigby et al. (2019). [Note that the above parameterization was used by Dean, Lawless and Willmot(1989). It is also a special case of the Sichel distribution SI() when \nu=-\frac{1}{2}.]

The probability function of the Poisson-inverse Gaussian distribution PIG2, see Heller, Couturier and Heritier (2018), is given by

f(y|\mu,\sigma)=\left( \frac{2 \sigma}{\pi}^{\frac{1}{2}}\right) \frac{\mu^y e^{\frac{1}{\sigma}} K_{y-\frac{1}{2}}(\sigma)}{(\alpha \sigma)^y y!}

for y=0,1,2,...,\infty, \mu>0 and \sigma>0 and \alpha = \left[(\mu^2+\sigma^2)^{0.5}-\mu \right]^{-1}, K_{\lambda}(t)=\frac{1}{2}\int_0^{\infty} x^{\lambda-1} \exp\{-\frac{1}{2}t(x+x^{-1})\}dx is the modified Bessel function of the third kind, see pp. 487-489 of Rigby et al. (2019).

The definition of the zero adjusted Poison inverse Gaussian distribution, ZAPIG and the the zero inflated Poison inverse Gaussian distribution, ZIPIG, are given in p. 513 and pp. 514-515 of of Rigby et al. (2019), respectively.

Value

Returns a gamlss.family object which can be used to fit a Poisson-inverse Gaussian distribution in the gamlss() function.

Author(s)

Dominique-Laurent Couturier, Mikis Stasinopoulos, Bob Rigby and Marco Enea

References

Dean, C., Lawless, J. F. and Willmot, G. E., A mixed poisson-inverse-Gaussian regression model, Canadian J. Statist., 17, 2, pp 171-181

Heller, G. Z., Couturier, D.L. and Heritier, S. R. (2018) Beyond mean modelling: Bias due to misspecification of dispersion in Poisson-inverse Gaussian regression Biometrical Journal, 2, pp 333-342.

Rigby, R. A. and Stasinopoulos D. M. (2005). Generalized additive models for location, scale and shape,(with discussion), Appl. Statist., 54, part 3, pp 507-554.

Rigby, R. A., Stasinopoulos, D. M., Heller, G. Z., and De Bastiani, F. (2019) Distributions for modeling location, scale, and shape: Using GAMLSS in R, Chapman and Hall/CRC, doi:10.1201/9780429298547. An older version can be found in https://www.gamlss.com/.

Stasinopoulos D. M. Rigby R.A. (2007) Generalized additive models for location scale and shape (GAMLSS) in R. Journal of Statistical Software, Vol. 23, Issue 7, Dec 2007, doi:10.18637/jss.v023.i07.

Stasinopoulos D. M., Rigby R.A., Heller G., Voudouris V., and De Bastiani F., (2017) Flexible Regression and Smoothing: Using GAMLSS in R, Chapman and Hall/CRC. doi:10.1201/b21973

(see also https://www.gamlss.com/).

See Also

gamlss.family, NBI, NBII, SI, SICHEL

Examples

PIG()# gives information about the default links for the  Poisson-inverse Gaussian distribution 
#plot the pdf using plot 
plot(function(y) dPIG(y, mu=10, sigma = 1 ), from=0, to=50, n=50+1, type="h") # pdf
# plot the cdf
plot(seq(from=0,to=50),pPIG(seq(from=0,to=50), mu=10, sigma=1), type="h")   # cdf
# generate random sample
tN <- table(Ni <- rPIG(100, mu=5, sigma=1))
r <- barplot(tN, col='lightblue')
# fit a model to the data 
# library(gamlss)
# gamlss(Ni~1,family=PIG)
ZIPIG()
ZAPIG()

Poisson distribution for fitting a GAMLSS model

Description

This function PO defines the Poisson distribution, an one parameter distribution, for a gamlss.family object to be used in GAMLSS fitting using the function gamlss(). The functions dPO, pPO, qPO and rPO define the density, distribution function, quantile function and random generation for the Poisson, PO(), distribution.

Usage

PO(mu.link = "log")
dPO(x, mu = 1, log = FALSE)
pPO(q, mu = 1, lower.tail = TRUE, log.p = FALSE)
qPO(p, mu = 1, lower.tail = TRUE, log.p = FALSE)
rPO(n, mu = 1)

Arguments

Details

Definition file for Poisson distribution.

f(y|\mu)=\frac{e^{-\mu}\mu^y}{\Gamma(y+1)}

for y=0,1,2,... and \mu>0 see ee pp 476-477 of Rigby et al. (2019).

Value

returns a gamlss.family object which can be used to fit a Poisson distribution in the gamlss() function.

Note

\mu is the mean of the Poisson distribution

Author(s)

Bob Rigby, Mikis Stasinopoulos, and Kalliope Akantziliotou

References

Rigby, R. A. and Stasinopoulos D. M. (2005). Generalized additive models for location, scale and shape,(with discussion), Appl. Statist., 54, part 3, pp 507-554.

Rigby, R. A., Stasinopoulos, D. M., Heller, G. Z., and De Bastiani, F. (2019) Distributions for modeling location, scale, and shape: Using GAMLSS in R, Chapman and Hall/CRC, doi:10.1201/9780429298547. An older version can be found in https://www.gamlss.com/.

Stasinopoulos D. M. Rigby R.A. (2007) Generalized additive models for location scale and shape (GAMLSS) in R. Journal of Statistical Software, Vol. 23, Issue 7, Dec 2007, doi:10.18637/jss.v023.i07.

(Stasinopoulos D. M., Rigby R.A., Heller G., Voudouris V., and De Bastiani F., (2017) Flexible Regression and Smoothing: Using GAMLSS in R, Chapman and Hall/CRC. doi:10.1201/b21973

(see also https://www.gamlss.com/).

See Also

gamlss.family, NBI, NBII, SI, SICHEL

Examples

PO()# gives information about the default links for the Poisson distribution  
# fitting data using PO()

# plotting the distribution
plot(function(y) dPO(y, mu=10 ), from=0, to=20, n=20+1, type="h")
# creating random variables and plot them 
tN <- table(Ni <- rPO(1000, mu=5))
 r <- barplot(tN, col='lightblue')
# library(gamlss)
# data(aids)
# h<-gamlss(y~cs(x,df=7)+qrt, family=PO, data=aids) # fits the constant+x+qrt model 
# plot(h)
# pdf.plot(family=PO, mu=10, min=0, max=20, step=1)

The Reverse Gumbel distribution for fitting a GAMLSS

Description

The function RG defines the reverse Gumbel distribution, a two parameter distribution, for a gamlss.family object to be used in GAMLSS fitting using the function gamlss(). The functions dRG, pRG, qRG and rRG define the density, distribution function, quantile function and random generation for the specific parameterization of the reverse Gumbel distribution.

Usage

RG(mu.link = "identity", sigma.link = "log")
dRG(x, mu = 0, sigma = 1, log = FALSE)
pRG(q, mu = 0, sigma = 1, lower.tail = TRUE, log.p = FALSE)
qRG(p, mu = 0, sigma = 1, lower.tail = TRUE, log.p = FALSE)
rRG(n, mu = 0, sigma = 1)

Arguments

Details

The specific parameterization of the reverse Gumbel distribution used in RG is

f(y|\mu,\sigma)= \frac{1}{\sigma} \hspace{1mm} \exp\left\{-\left(\frac{y-\mu}{\sigma}\right)-\exp\left[-\left(\frac{y-\mu)}{\sigma}\right)\right]\right\}

for y=(-\infty,\infty), \mu=(-\infty,+\infty) and \sigma>0 see pp. 370-371 of Rigby et al. (2019).

Value

RG() returns a gamlss.family object which can be used to fit a Gumbel distribution in the gamlss() function. dRG() gives the density, pGU() gives the distribution function, qRG() gives the quantile function, and rRG() generates random deviates.

Note

The mean of the distribution is \mu+0.57722 \sigma and the variance is \pi^2 \sigma^2/6.

Author(s)

Mikis Stasinopoulos, Bob Rigby and Calliope Akantziliotou

References

Rigby, R. A. and Stasinopoulos D. M. (2005). Generalized additive models for location, scale and shape,(with discussion), Appl. Statist., 54, part 3, pp 507-554.

Rigby, R. A., Stasinopoulos, D. M., Heller, G. Z., and De Bastiani, F. (2019) Distributions for modeling location, scale, and shape: Using GAMLSS in R, Chapman and Hall/CRC, doi:10.1201/9780429298547. An older version can be found in https://www.gamlss.com/.

Stasinopoulos D. M. Rigby R.A. (2007) Generalized additive models for location scale and shape (GAMLSS) in R. Journal of Statistical Software, Vol. 23, Issue 7, Dec 2007, doi:10.18637/jss.v023.i07.

Stasinopoulos D. M., Rigby R.A., Heller G., Voudouris V., and De Bastiani F., (2017) Flexible Regression and Smoothing: Using GAMLSS in R, Chapman and Hall/CRC. doi:10.1201/b21973

(see also https://www.gamlss.com/).

See Also

gamlss.family

Examples

plot(function(x) dRG(x, mu=0,sigma=1), -3, 6, 
 main = "{Reverse Gumbel  density mu=0,sigma=1}")
RG()# gives information about the default links for the Gumbel distribution      
dat<-rRG(100, mu=10, sigma=2) # generates 100 random observations 
# library(gamlss)
# gamlss(dat~1,family=RG) # fits a constant for each parameter mu and sigma 

Reverse generalized extreme family distribution for fitting a GAMLSS

Description

The function RGE defines the reverse generalized extreme family distribution, a three parameter distribution, for a gamlss.family object to be used in GAMLSS fitting using the function gamlss(). The functions dRGE, pRGE, qRGE and rRGE define the density, distribution function, quantile function and random generation for the specific parameterization of the reverse generalized extreme distribution given in details below.

Usage

RGE(mu.link = "identity", sigma.link = "log", nu.link = "log")
dRGE(x, mu = 1, sigma = 0.1, nu = 1, log = FALSE)
pRGE(q, mu = 1, sigma = 0.1, nu = 1, lower.tail = TRUE, log.p = FALSE)
qRGE(p, mu = 1, sigma = 0.1, nu = 1, lower.tail = TRUE, log.p = FALSE) 
rRGE(n, mu = 1, sigma = 0.1, nu = 1)

Arguments

Details

Definition file for reverse generalized extreme family distribution.

The probability density function of the generalized extreme value distribution is obtained from Johnson et al. (1995), Volume 2, p76, equation (22.184) [where (\xi, \theta, \gamma) \longrightarrow (\mu, \sigma, \nu)].

The probability density function of the reverse generalized extreme value distribution is then obtained by replacing y by -y and \mu by -\mu.

Hence the probability density function of the reverse generalized extreme value distribution with \nu>0 is given by

f(y|\mu,\sigma, \nu)=\frac{1}{\sigma}\left[1+\frac{\nu(y-\mu)}{\sigma}\right]^{\frac{1}{\nu}-1}S_1(y|\mu,\sigma,\nu)

for

\mu-\frac{\sigma}{\nu}<y<\infty

where

S_1(y|\mu,\sigma,\nu)=\exp\left\{-\left[1+\frac{\nu(y-\mu)}{\sigma}\right]^\frac{1}{\nu}\right\}

and where -\infty<\mu<y+\frac{\sigma}{\nu}, \sigma>0 and \nu>0. Note that only the case \nu>0 is allowed here. The reverse generalized extreme value distribution is denoted as RGE(\mu,\sigma,\nu) or as Reverse Generalized.Extreme.Family(\mu,\sigma,\nu).

Note the the above distribution is a reparameterization of the three parameter Weibull distribution given by

f(y|\alpha_1,\alpha_2,\alpha_3)=\frac{\alpha_3}{\alpha_2}\left[\frac{y-\alpha_1}{\alpha_2}\right]^{\alpha_3-1} \exp\left[ -\left(\frac{y-\alpha_1}{\alpha_2} \right)^{\alpha_3} \right]

given by setting \alpha_1=\mu-\sigma/\nu, \alpha_2=\sigma/\nu, \alpha_3=1/\nu.

Value

RGE() returns a gamlss.family object which can be used to fit a reverse generalized extreme distribution in the gamlss() function. dRGE() gives the density, pRGE() gives the distribution function, qRGE() gives the quantile function, and rRGE() generates random deviates.

Note

This distribution is very difficult to fit because the y values depends on the parameter values. The RS() and CG() algorithms are not appropriate for this type of problem.

Author(s)

Bob Rigby, Mikis Stasinopoulos and Kalliope Akantziliotou

References

Rigby, R. A. and Stasinopoulos D. M. (2005). Generalized additive models for location, scale and shape,(with discussion), Appl. Statist., 54, part 3, pp 507-554.

Rigby, R. A., Stasinopoulos, D. M., Heller, G. Z., and De Bastiani, F. (2019) Distributions for modeling location, scale, and shape: Using GAMLSS in R, Chapman and Hall/CRC, doi:10.1201/9780429298547. An older version can be found in https://www.gamlss.com/.

Stasinopoulos D. M. Rigby R.A. (2007) Generalized additive models for location scale and shape (GAMLSS) in R. Journal of Statistical Software, Vol. 23, Issue 7, Dec 2007, doi:10.18637/jss.v023.i07.

Stasinopoulos D. M., Rigby R.A., Heller G., Voudouris V., and De Bastiani F., (2017) Flexible Regression and Smoothing: Using GAMLSS in R, Chapman and Hall/CRC. doi:10.1201/b21973

(see also https://www.gamlss.com/).

See Also

gamlss.family

Examples

RGE()# default links for the reverse generalized extreme family distribution 
newdata<-rRGE(100,mu=0,sigma=1,nu=5) # generates 100 random observations
# library(gamlss)
# gamlss(newdata~1, family=RGE, method=mixed(5,50)) # difficult to converse 

The Skew Power exponential (SEP) distribution for fitting a GAMLSS

Description

This function defines the Skew Power exponential (SEP) distribution, a four parameter distribution, for a gamlss.family object to be used for a GAMLSS fitting using the function gamlss(). The functions dSEP, pSEP, qSEP and rSEP define the density, distribution function, quantile function and random generation for the Skew Power exponential (SEP) distribution.

Usage

SEP(mu.link = "identity", sigma.link = "log", nu.link = "identity", 
    tau.link = "log")
dSEP(x, mu = 0, sigma = 1, nu = 0, tau = 2, log = FALSE)
pSEP(q, mu = 0, sigma = 1, nu = 0, tau = 2, lower.tail = TRUE, 
     log.p = FALSE)
qSEP(p, mu = 0, sigma = 1, nu = 0, tau = 2, lower.tail = TRUE, 
     log.p = FALSE, lower.limit = mu - 5 * sigma, 
     upper.limit = mu + 5 * sigma)
rSEP(n, mu = 0, sigma = 1, nu = 0, tau = 2)

Arguments

Details

The probability density function of the Skew Power exponential distribution, (SEP), is defined as

f(y|n,\mu,\sigma\,\nu,\tau)==\frac{z}{\sigma} \Phi(\omega) \hspace{1mm} f_{EP}(z,0,1,\tau)

for -\infty < y < \infty , \mu=(-\infty,+\infty), \sigma>0, \nu=(-\infty,+\infty) and \tau>0. where z = \frac{y-\mu}{\sigma}, \omega = sign(z)|z|^{\tau/2}\nu \sqrt{2/\tau} and f_{EP}(z,0,1,\tau) is the pdf of an Exponential Power distribution.

Value

SEP() returns a gamlss.family object which can be used to fit the SEP distribution in the gamlss() function. dSEP() gives the density, pSEP() gives the distribution function, qSEP() gives the quantile function, and rSEP() generates random deviates.

Warning

The qSEP and rSEP are slow since they are relying on golden section for finding the quantiles

Author(s)

Bob Rigby and Mikis Stasinopoulos

References

Diciccio, T. J. and Mondi A. C. (2004). Inferential Aspects of the Skew Exponential Power distribution., JASA, 99, 439-450.

Rigby, R. A. and Stasinopoulos D. M. (2005). Generalized additive models for location, scale and shape,(with discussion), Appl. Statist., 54, part 3, pp 507-554.

Rigby, R. A., Stasinopoulos, D. M., Heller, G. Z., and De Bastiani, F. (2019) Distributions for modeling location, scale, and shape: Using GAMLSS in R, Chapman and Hall/CRC, doi:10.1201/9780429298547. An older version can be found in https://www.gamlss.com/.

Stasinopoulos D. M. Rigby R.A. (2007) Generalized additive models for location scale and shape (GAMLSS) in R. Journal of Statistical Software, Vol. 23, Issue 7, Dec 2007, doi:10.18637/jss.v023.i07.

Stasinopoulos D. M., Rigby R.A., Heller G., Voudouris V., and De Bastiani F., (2017) Stasinopoulos D. M., Rigby R.A., Heller G., Voudouris V., and De Bastiani F., (2017) Flexible Regression and Smoothing: Using GAMLSS in R, Chapman and Hall/CRC. doi:10.1201/b21973

(see also https://www.gamlss.com/).

See Also

gamlss.family, JSU, BCT

Examples

SEP()   # 
plot(function(x)dSEP(x, mu=0,sigma=1, nu=1, tau=2), -5, 5, 
 main = "The SEP  density mu=0,sigma=1,nu=1, tau=2")
plot(function(x) pSEP(x, mu=0,sigma=1,nu=1, tau=2), -5, 5, 
 main = "The BCPE  cdf mu=0, sigma=1, nu=1, tau=2")
dat <- rSEP(100,mu=10,sigma=1,nu=-1,tau=1.5)
# library(gamlss)
# gamlss(dat~1,family=SEP, control=gamlss.control(n.cyc=30))

The Skew exponential power type 1-4 distribution for fitting a GAMLSS

Description

These functions define the Skew Power exponential type 1 to 4 distributions. All of them are four parameter distributions and can be used to fit a GAMLSS model. The functions dSEP1, dSEP2, dSEP3 and dSEP4 define the probability distribution functions, the functions pSEP1, pSEP2, pSEP3 and pSEP4 define the cumulative distribution functions the functions qSEP1, qSEP2, qSEP3 and qSEP4 define the inverse cumulative distribution functions and the functions rSEP1, rSEP2, rSEP3 and rSEP4 define the random generation for the Skew exponential power distributions.

Usage

SEP1(mu.link = "identity", sigma.link = "log", nu.link = "identity", 
     tau.link = "log")
dSEP1(x, mu = 0, sigma = 1, nu = 0, tau = 2, log = FALSE)
pSEP1(q, mu = 0, sigma = 1, nu = 0, tau = 2, lower.tail = TRUE, 
     log.p = FALSE)
qSEP1(p, mu = 0, sigma = 1, nu = 0, tau = 2, lower.tail = TRUE, 
     log.p = FALSE)
rSEP1(n, mu = 0, sigma = 1, nu = 0, tau = 2)

SEP2(mu.link = "identity", sigma.link = "log", nu.link = "identity", 
      tau.link = "log")
dSEP2(x, mu = 0, sigma = 1, nu = 0, tau = 2, log = FALSE)
pSEP2(q, mu = 0, sigma = 1, nu = 0, tau = 2, lower.tail = TRUE, 
      log.p = FALSE)
qSEP2(p, mu = 0, sigma = 1, nu = 0, tau = 2, lower.tail = TRUE, 
      log.p = FALSE)
rSEP2(n, mu = 0, sigma = 1, nu = 0, tau = 2)

SEP3(mu.link = "identity", sigma.link = "log", nu.link = "log", 
      tau.link = "log")
dSEP3(x, mu = 0, sigma = 1, nu = 2, tau = 2, log = FALSE)
pSEP3(q, mu = 0, sigma = 1, nu = 2, tau = 2, lower.tail = TRUE, 
      log.p = FALSE)
qSEP3(p, mu = 0, sigma = 1, nu = 2, tau = 2, lower.tail = TRUE, 
      log.p = FALSE)

SEP4(mu.link = "identity", sigma.link = "log", nu.link = "log", 
      tau.link = "log")
dSEP4(x, mu = 0, sigma = 1, nu = 2, tau = 2, log = FALSE)      
pSEP4(q, mu = 0, sigma = 1, nu = 2, tau = 2, lower.tail = TRUE, 
      log.p = FALSE)
qSEP4(p, mu = 0, sigma = 1, nu = 2, tau = 2, lower.tail = TRUE, 
      log.p = FALSE)
rSEP4(n, mu = 0, sigma = 1, nu = 2, tau = 2)

Arguments

Details

The probability density function of the Skew Power exponential distribution type 1, (SEP1), is defined as

f_Y(y|\mu,\sigma\,\nu,\tau)=\frac{2}{\sigma} f_{Z_1}(z) F_{Z_1}(\nu z)

for -\infty < y < \infty , -\infty<\mu <\infty, \sigma>0, -\infty<\nu <\infty, and \tau>0 where z=(y-\mu)/\sigma, and Z_1 \sim \texttt{PE2}(0, \tau^{1/\tau }, \tau) , see pp 401-402 of Rigby et al. (2019).

The probability density function of the Skew Power exponential distribution type 2, (SEP2), is defined as

f_Y(y|\mu,\sigma\,\nu,\tau)=\frac{2}{\sigma} f_{Z_1}(z) \Phi_{Z_1}(\omega)

for -\infty < y < \infty , -\infty<\mu <\infty, \sigma>0, -\infty<\nu <\infty, and \tau>0 where z=(y-\mu)/\sigma, and \omega= \texttt{sign}(z) |z|^{\tau/2} \nu \sqrt{2/\tau}, and Z_1 \sim \texttt{PE2}(0, \tau^{1/\tau }, \tau) , see pp 402-404 of Rigby et al. (2019).

For SEP3 and SEP3 see pp 404-406 and pp 407-408 of Rigby et al. (2019), respectively.

Value

SEP2() returns a gamlss.family object which can be used to fit the SEP2 distribution in the gamlss() function. dSEP2() gives the density, pSEP2() gives the distribution function, qSEP2() gives the quantile function, and rSEP2() generates random deviates.

Author(s)

Bob Rigby and Mikis Stasinopoulos

References

Fernadez C., Osiewalski J. and Steel M.F.J.(1995) Modelling and inference with v-spherical distributions. JASA, 90, pp 1331-1340.

Rigby, R. A. and Stasinopoulos D. M. (2005). Generalized additive models for location, scale and shape,(with discussion), Appl. Statist., 54, part 3, pp 507-554.

Rigby, R. A., Stasinopoulos, D. M., Heller, G. Z., and De Bastiani, F. (2019) Distributions for modeling location, scale, and shape: Using GAMLSS in R, Chapman and Hall/CRC, doi:10.1201/9780429298547. An older version can be found in https://www.gamlss.com/.

Stasinopoulos D. M. Rigby R.A. (2007) Generalized additive models for location scale and shape (GAMLSS) in R. Journal of Statistical Software, Vol. 23, Issue 7, Dec 2007, doi:10.18637/jss.v023.i07.

Stasinopoulos D. M., Rigby R.A., Heller G., Voudouris V., and De Bastiani F., (2017) Flexible Regression and Smoothing: Using GAMLSS in R, Chapman and Hall/CRC. doi:10.1201/b21973

(see also https://www.gamlss.com/).

See Also

gamlss.family, SEP

Examples

SEP1() 
curve(dSEP4(x, mu=5 ,sigma=1, nu=2, tau=1.5), -2, 10, 
          main = "The SEP4  density mu=5 ,sigma=1, nu=1, tau=1.5")
# library(gamlss)
#y<- rSEP4(100, mu=5, sigma=1, nu=2, tau=1.5);hist(y)
#m1<-gamlss(y~1, family=SEP1, n.cyc=50)
#m2<-gamlss(y~1, family=SEP2, n.cyc=50)
#m3<-gamlss(y~1, family=SEP3, n.cyc=50)
#m4<-gamlss(y~1, family=SEP4, n.cyc=50) 
#GAIC(m1,m2,m3,m4)

The Sinh-Arcsinh (SHASH) distribution for fitting a GAMLSS

Description

The Sinh-Arcsinh (SHASH) distribution is a four parameter distribution, for a gamlss.family object to be used for a GAMLSS fitting using the function gamlss(). The functions dSHASH, pSHASH, qSHASH and rSHASH define the density, distribution function, quantile function and random generation for the Sinh-Arcsinh (SHASH) distribution.

There are 3 different SHASH distributions implemented in GAMLSS.

Usage

SHASH(mu.link = "identity", sigma.link = "log", nu.link = "log", 
      tau.link = "log")
dSHASH(x, mu = 0, sigma = 1, nu = 0.5, tau = 0.5, log = FALSE)
pSHASH(q, mu = 0, sigma = 1, nu = 0.5, tau = 0.5, lower.tail = TRUE, 
     log.p = FALSE)
qSHASH(p, mu = 0, sigma = 1, nu = 0.5, tau = 0.5, lower.tail = TRUE, 
     log.p = FALSE)
rSHASH(n, mu = 0, sigma = 1, nu = 0.5, tau = 0.5)

SHASHo(mu.link = "identity", sigma.link = "log", nu.link = "identity", 
      tau.link = "log")
dSHASHo(x, mu = 0, sigma = 1, nu = 0, tau = 1, log = FALSE)
pSHASHo(q, mu = 0, sigma = 1, nu = 0, tau = 1, lower.tail = TRUE, 
     log.p = FALSE)
qSHASHo(p, mu = 0, sigma = 1, nu = 0, tau = 1, lower.tail = TRUE, 
     log.p = FALSE)
rSHASHo(n, mu = 0, sigma = 1, nu = 0, tau = 1)

SHASHo2(mu.link = "identity", sigma.link = "log", nu.link = "identity", 
      tau.link = "log")
dSHASHo2(x, mu = 0, sigma = 1, nu = 0, tau = 1, log = FALSE)
pSHASHo2(q, mu = 0, sigma = 1, nu = 0, tau = 1, lower.tail = TRUE, 
     log.p = FALSE)
qSHASHo2(p, mu = 0, sigma = 1, nu = 0, tau = 1, lower.tail = TRUE, 
     log.p = FALSE)
rSHASHo2(n, mu = 0, sigma = 1, nu = 0, tau = 1)

Arguments

Details

The probability density function of the Sinh-Arcsinh distribution, SHASH, Jones(2005), is defined as

f(y|\mu,\sigma\,\nu,\tau) = \frac{c}{\sqrt{2 \pi} \sigma (1+z^2)^{1/2}} e^{-r^2/2}

where

r=\frac{1}{2} \left \{ \exp\left[ \tau \sinh^{-1}(z) \right] -\exp\left[ -\nu \sinh^{-1}(z) \right] \right\}