Type: | Package |
Title: | The Asymmetric Laplace Distribution |
Version: | 1.3.1 |
Date: | 2021-04-04 |
Author: | Christian E. Galarza <cgalarza88@gmail.com> and Victor H. Lachos <hlachos@ime.unicamp.br> |
Maintainer: | Christian E. Galarza <cgalarza88@gmail.com> |
Description: | It provides the density, distribution function, quantile function, random number generator, likelihood function, moments and Maximum Likelihood estimators for a given sample, all this for the three parameter Asymmetric Laplace Distribution defined in Koenker and Machado (1999). This is a special case of the skewed family of distributions available in Galarza et.al. (2017) <doi:10.1002/sta4.140> useful for quantile regression. |
License: | GPL-2 | GPL-3 [expanded from: GPL (≥ 2)] |
NeedsCompilation: | no |
Packaged: | 2021-04-04 18:49:07 UTC; cgala |
Repository: | CRAN |
Date/Publication: | 2021-04-04 19:10:02 UTC |
The Asymmetric Laplace Distribution
Description
It provides the density, distribution function, quantile function, random number generator, likelihood function, moments and Maximum Likelihood estimators for a given sample, all this for the three parameter Asymmetric Laplace Distribution defined in Koenker and Machado (1999) useful for quantile regression.
Details
Package: | ald |
Type: | Package |
Version: | 1.0 |
Date: | 2015-01-27 |
License: | GPL (>=2) |
Author(s)
Christian E. Galarza <cgalarza88@gmail.com> and Victor H. Lachos <hlachos@ime.unicamp.br>
References
Koenker, R., Machado, J. (1999). Goodness of fit and related inference processes for quantile regression. J. Amer. Statist. Assoc. 94(3):1296-1309.
Yu, K. & Moyeed, R. (2001). Bayesian quantile regression. Statistics & Probability Letters, 54(4), 437-447.
Yu, K., & Zhang, J. (2005). A three-parameter asymmetric Laplace distribution and its extension. Communications in Statistics-Theory and Methods, 34(9-10), 1867-1879.
See Also
Examples
## Let's plot an Asymmetric Laplace Distribution!
##Density
sseq = seq(-40,80,0.5)
dens = dALD(y=sseq,mu=50,sigma=3,p=0.75)
plot(sseq,dens,type="l",lwd=2,col="red",xlab="x",ylab="f(x)", main="ALD Density function")
## Distribution Function
df = pALD(q=sseq,mu=50,sigma=3,p=0.75)
plot(sseq,df,type="l",lwd=2,col="blue",xlab="x",ylab="F(x)", main="ALD Distribution function")
abline(h=1,lty=2)
##Inverse Distribution Function
prob = seq(0,1,length.out = 1000)
idf = qALD(prob=prob,mu=50,sigma=3,p=0.75)
plot(prob,idf,type="l",lwd=2,col="gray30",xlab="x",ylab=expression(F^{-1}~(x)))
title(main="ALD Inverse Distribution function")
abline(v=c(0,1),lty=2)
#Random Sample Histogram
sample = rALD(n=10000,mu=50,sigma=3,p=0.75)
hist(sample,breaks = 70,freq = FALSE,ylim=c(0,max(dens)),main="")
title(main="Histogram and True density")
lines(sseq,dens,col="red",lwd=2)
## Let's compute the MLE's
param = c(-323,40,0.9)
y = rALD(10000,mu = param[1],sigma = param[2],p = param[3]) #A random sample
res = mleALD(y)
#Comparing
cbind(param,res$par)
#Let's plot
seqq = seq(min(y),max(y),length.out = 1000)
dens = dALD(y=seqq,mu=res$par[1],sigma=res$par[2],p=res$par[3])
hist(y,breaks=50,freq = FALSE,ylim=c(0,max(dens)))
lines(seqq,dens,type="l",lwd=2,col="red",xlab="x",ylab="f(x)", main="ALD Density function")
The Asymmetric Laplace Distribution
Description
Density, distribution function, quantile function and random generation for a Three-Parameter Asymmetric Laplace Distribution defined in Koenker and Machado (1999) useful for quantile regression with location parameter equal to mu
, scale parameter sigma
and skewness parameter p
This is a special case of the skewed family of distributions in Galarza (2016) available in lqr::SKD
.
Usage
dALD(y, mu = 0, sigma = 1, p = 0.5)
pALD(q, mu = 0, sigma = 1, p = 0.5, lower.tail = TRUE)
qALD(prob, mu = 0, sigma = 1, p = 0.5, lower.tail = TRUE)
rALD(n, mu = 0, sigma = 1, p = 0.5)
Arguments
y , q |
vector of quantiles. |
prob |
vector of probabilities. |
n |
number of observations. |
mu |
location parameter. |
sigma |
scale parameter. |
p |
skewness parameter. |
lower.tail |
logical; if TRUE (default), probabilities are P[X |
Details
If mu
, sigma
or p
are not specified they assume the default values of 0, 1 and 0.5, respectively, belonging to the Symmetric Standard Laplace Distribution denoted by ALD(0,1,0.5)
.
As discussed in Koenker and Machado (1999) and Yu and Moyeed (2001) we say that a random variable
Y is distributed as an ALD with location parameter \mu
, scale parameter \sigma>0
and skewness parameter p
in (0,1), if its probability density function (pdf) is given by
f(y|\mu,\sigma,p)=\frac{p(1-p)}{\sigma}\exp
{-\rho_{p}(\frac{y-\mu}{\sigma})}
where \rho_p(.)
is the so called check (or loss) function defined by
\rho_p(u)=u(p - I_{u<0})
,
with I_{.}
denoting the usual indicator function. This distribution is denoted by ALD(\mu,\sigma,p)
and it's p
-th quantile is equal to \mu
.
The scale parameter sigma
must be positive and non zero. The skew parameter p
must be between zero and one (0<p
<1).
Value
dALD
gives the density, pALD
gives the distribution function, qALD
gives the quantile function, and rALD
generates a random sample.
The length of the result is determined by n for rALD
, and is the maximum of the lengths of the numerical arguments for the other functions dALD
, pALD
and qALD
.
Note
The numerical arguments other than n
are recycled to the length of the result.
Author(s)
Christian E. Galarza <cgalarza88@gmail.com> and Victor H. Lachos <hlachos@ime.unicamp.br>
References
Galarza Morales, C., Lachos Davila, V., Barbosa Cabral, C., and Castro Cepero, L. (2017) Robust quantile regression using a generalized class of skewed distributions. Stat,6: 113-130 doi: 10.1002/sta4.140.
Yu, K., & Zhang, J. (2005). A three-parameter asymmetric Laplace distribution and its extension. Communications in Statistics-Theory and Methods, 34(9-10), 1867-1879.
See Also
Examples
## Let's plot an Asymmetric Laplace Distribution!
##Density
library(ald)
sseq = seq(-40,80,0.5)
dens = dALD(y=sseq,mu=50,sigma=3,p=0.75)
plot(sseq,dens,type = "l",lwd=2,col="red",xlab="x",ylab="f(x)", main="ALD Density function")
#Look that is a special case of the skewed family in Galarza (2017)
# available in lqr package, dSKD(...,sigma = 2*3,dist = "laplace")
## Distribution Function
df = pALD(q=sseq,mu=50,sigma=3,p=0.75)
plot(sseq,df,type="l",lwd=2,col="blue",xlab="x",ylab="F(x)", main="ALD Distribution function")
abline(h=1,lty=2)
##Inverse Distribution Function
prob = seq(0,1,length.out = 1000)
idf = qALD(prob=prob,mu=50,sigma=3,p=0.75)
plot(prob,idf,type="l",lwd=2,col="gray30",xlab="x",ylab=expression(F^{-1}~(x)))
title(main="ALD Inverse Distribution function")
abline(v=c(0,1),lty=2)
#Random Sample Histogram
sample = rALD(n=10000,mu=50,sigma=3,p=0.75)
hist(sample,breaks = 70,freq = FALSE,ylim=c(0,max(dens)),main="")
title(main="Histogram and True density")
lines(sseq,dens,col="red",lwd=2)
Log-Likelihood function for the Asymmetric Laplace Distribution
Description
Log-Likelihood function for the Three-Parameter Asymmetric Laplace Distribution defined in Koenker and Machado (1999) useful for quantile regression with location parameter equal to mu
, scale parameter sigma
and skewness parameter p
.
Usage
likALD(y, mu = 0, sigma = 1, p = 0.5, loglik = TRUE)
Arguments
y |
observation vector. |
mu |
location parameter |
sigma |
scale parameter |
p |
skewness parameter |
loglik |
logical; if TRUE (default), the Log-likelihood is return, if not just the Likelihood. |
Details
If mu
, sigma
or p
are not specified they assume the default values of 0, 1 and 0.5, respectively, belonging to the Symmetric Standard Laplace Distribution denoted by ALD(0,1,0.5)
.
As discussed in Koenker and Machado (1999) and Yu and Moyeed (2001) we say that a random variable
Y is distributed as an ALD with location parameter \mu
, scale parameter \sigma>0
and skewness parameter p
in (0,1), if its probability density function (pdf) is given by
f(y|\mu,\sigma,p)=\frac{p(1-p)}{\sigma}\exp
{-\rho_{p}(\frac{y-\mu}{\sigma})}
where \rho_p(.)
is the so called check (or loss) function defined by
\rho_p(u)=u(p - I_{u<0})
,
with I_{.}
denoting the usual indicator function. Then the Log-likelihood function is given by
\sum_{i=1}^{n}log(\frac{p(1-p)}{\sigma}\exp
{-\rho_{p}(\frac{y_i-\mu}{\sigma})})
.
The scale parameter sigma
must be positive and non zero. The skew parameter p
must be between zero and one (0<p
<1).
Value
likeALD
returns the Log-likelihood by default and just the Likelihood if loglik = FALSE
.
Author(s)
Christian E. Galarza <cgalarza88@gmail.com> and Victor H. Lachos <hlachos@ime.unicamp.br>
References
Koenker, R., Machado, J. (1999). Goodness of fit and related inference processes for quantile regression. J. Amer. Statist. Assoc. 94(3):1296-1309.
Yu, K. & Moyeed, R. (2001). Bayesian quantile regression. Statistics & Probability Letters, 54(4), 437-447.
Yu, K., & Zhang, J. (2005). A three-parameter asymmetric Laplace distribution and its extension. Communications in Statistics-Theory and Methods, 34(9-10), 1867-1879.
See Also
Examples
## Let's compute the log-likelihood for a given sample
y = rALD(n=1000)
loglik = likALD(y)
#Changing the true parameters the loglik must decrease
loglik2 = likALD(y,mu=10,sigma=2,p=0.3)
loglik;loglik2
if(loglik>loglik2){print("First parameters are Better")}
Maximum Likelihood Estimators (MLE) for the Asymmetric Laplace Distribution
Description
Maximum Likelihood Estimators (MLE) for the Three-Parameter Asymmetric Laplace Distribution defined in Koenker and Machado (1999) useful for quantile regression with location parameter equal to mu
, scale parameter sigma
and skewness parameter p
.
Usage
mleALD(y, initial = NA)
Arguments
y |
observation vector. |
initial |
optional vector of initial values c( |
Details
The algorithm computes iteratevely the MLE's via the combination of the MLE expressions for \mu
and \sigma
, and then maximizing with rescpect to p
the Log-likelihood function (likALD
) using the well known optimize
R function. By default the tolerance is 10^-5 for all parameters.
Value
The function returns a list with two objects
iter |
iterations to reach convergence. |
par |
vector of Maximum Likelihood Estimators. |
Author(s)
Christian E. Galarza <cgalarza88@gmail.com> and Victor H. Lachos <hlachos@ime.unicamp.br>
References
Yu, K., & Zhang, J. (2005). A three-parameter asymmetric Laplace distribution and its extension. Communications in Statistics-Theory and Methods, 34(9-10), 1867-1879.
See Also
Examples
## Let's try this function
param = c(-323,40,0.9)
y = rALD(10000,mu = param[1],sigma = param[2],p = param[3]) #A random sample
res = mleALD(y)
#Comparing
cbind(param,res$par)
#Let's plot
seqq = seq(min(y),max(y),length.out = 1000)
dens = dALD(y=seqq,mu=res$par[1],sigma=res$par[2],p=res$par[3])
hist(y,breaks=50,freq = FALSE,ylim=c(0,max(dens)))
lines(seqq,dens,type="l",lwd=2,col="red",xlab="x",ylab="f(x)", main="ALD Density function")
Moments for the Asymmetric Laplace Distribution
Description
Mean, variance, skewness, kurtosis, central moments w.r.t mu
and first absolute central moment for the Three-Parameter Asymmetric Laplace Distribution defined in Koenker and Machado (1999) useful for quantile regression with location parameter equal to mu
, scale parameter sigma
and skewness parameter p
.
Usage
meanALD(mu=0,sigma=1,p=0.5)
varALD(mu=0,sigma=1,p=0.5)
skewALD(mu=0,sigma=1,p=0.5)
kurtALD(mu=0,sigma=1,p=0.5)
momentALD(k=1,mu=0,sigma=1,p=0.5)
absALD(sigma=1,p=0.5)
Arguments
k |
moment number. |
mu |
location parameter |
sigma |
scale parameter |
p |
skewness parameter |
Details
If mu
, sigma
or p
are not specified they assume the default values of 0, 1 and 0.5, respectively, belonging to the Symmetric Standard Laplace Distribution denoted by ALD(0,1,0.5)
.
As discussed in Koenker and Machado (1999) and Yu and Moyeed (2001) we say that a random variable
Y is distributed as an ALD with location parameter \mu
, scale parameter \sigma>0
and skewness parameter p
in (0,1), if its probability density function (pdf) is given by
f(y|\mu,\sigma,p)=\frac{p(1-p)}{\sigma}\exp
{-\rho_{p}(\frac{y-\mu}{\sigma})}
where \rho_p(.)
is the so called check (or loss) function defined by
\rho_p(u)=u(p - I_{u<0})
,
with I_{.}
denoting the usual indicator function. This distribution is denoted by ALD(\mu,\sigma,p)
and it's p
th quantile is equal to \mu
. The scale parameter sigma
must be positive and non zero. The skew parameter p
must be between zero and one (0<p
<1).
Value
meanALD
gives the mean, varALD
gives the variance, skewALD
gives the skewness, kurtALD
gives the kurtosis, momentALD
gives the k
th central moment, i.e., E(y-\mu)^k
and absALD
gives the first absolute central moment denoted by E|y-\mu|
.
Author(s)
Christian E. Galarza <cgalarza88@gmail.com> and Victor H. Lachos <hlachos@ime.unicamp.br>
References
Koenker, R., Machado, J. (1999). Goodness of fit and related inference processes for quantile regression. J. Amer. Statist. Assoc. 94(3):1296-1309.
Yu, K. & Moyeed, R. (2001). Bayesian quantile regression. Statistics & Probability Letters, 54(4), 437-447.
Yu, K., & Zhang, J. (2005). A three-parameter asymmetric Laplace distribution and its extension. Communications in Statistics-Theory and Methods, 34(9-10), 1867-1879.
See Also
Examples
## Let's compute some moments for a Symmetric Standard Laplace Distribution.
#Third raw moment
momentALD(k=3,mu=0,sigma=1,p=0.5)
#The well known mean, variance, skewness and kurtosis
meanALD(mu=0,sigma=1,p=0.5)
varALD(mu=0,sigma=1,p=0.5)
skewALD(mu=0,sigma=1,p=0.5)
kurtALD(mu=0,sigma=1,p=0.5)
# and this guy
absALD(sigma=1,p=0.5)