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Type:

Package

Title:

Analysing Conditional Income Distributions

Version:

1.1

Date:

2016-02-01

Author:

Alexander Sohn

Maintainer:

Alexander Sohn <asohn@uni-goettingen.de>

Description:

Functions for the analysis of income distributions for subgroups of the population as defined by a set of variables like age, gender, region, etc. This entails a Kolmogorov-Smirnov test for a mixture distribution as well as functions for moments, inequality measures, entropy measures and polarisation measures of income distributions. This package thus aides the analysis of income inequality by offering tools for the exploratory analysis of income distributions at the disaggregated level.

Depends:

gamlss,gamlss.dist,Hmisc,stats,graphics,splines

Imports:

grDevices,utils,datasets,methods

Suggests:

ineq

License:

GPL-3

Packaged:

2016-02-01 06:36:11 UTC; admin

NeedsCompilation:

Repository:

CRAN

Date/Publication:

2016-02-01 13:34:52

Analysing Conditional Income Distributions

Description

Details

Package:	acid
Type:	Package
Version:	1.1
Date:	2015-01-06
License:	GPL-3

sadr.test, polarisation.ER, gini.den

Author(s)

Alexander Sohn <asohn@uni-goettingen.de>

References

Klein, N. and Kneib, T., Lang, S. and Sohn, A. (2015): Bayesian Structured Additive Distributional Regression with an Application to Regional Income Inequality in Germany, in: Annals of Applied Statistics, Vol. 9(2), pp. 1024-1052.

Sohn, A., Klein, N. and Kneib. T. (2014): A New Semiparametric Approach to Analysing Conditional Income Distributions, in: SOEPpapers, No. 676.

Mean of the Generalised Beta Distribution of Second Kind

Description

This function calculates the expectation of the Generalised Beta Distribution of Second Kind.

Usage

arithmean.GB2(b, a, p, q)

Arguments

b

the parameter b of the Dagum distribution as defined by Kleiber and Kotz (2003).

a

the parameter a of the Dagum distribution as defined by Kleiber and Kotz (2003).

p

the parameter p of the Dagum distribution as defined by Kleiber and Kotz (2003).

q

the parameter q of the Dagum distribution as defined by Kleiber and Kotz (2003).

Value

returns the expectation.

Author(s)

Alexander Sohn

References

Kleiber, C. and Kotz, S. (2003): Statistical Size Distributions in Economics and Actuarial Sciences, Wiley, Hoboken.

Examples

a.test<- 4
b.test<- 20000
p.test<- 0.7
q.test<- 1
alpha.test<-1
GB2sample<-rGB2(10000,b.test,a.test,p.test,q.test)
arithmean.GB2(b.test,a.test,p.test,q.test)
mean(GB2sample)

Atkinson Inequality Index

Description

This function computes the Atkinson inequality index for a vector of observations.

Usage

atkinson(x, epsilon = 1)

Arguments

x

a vector of observations.

epsilon

inequality aversion parameter as denoted by Atkinson (1970). The default is epsilon=1.

Value

returns the selected Atkinson inequality index.

Author(s)

Alexander Sohn

References

Atkinson, A.B. (1970): On the Measurment of Inequality, in: Journal of Economic Theory, Vol. 2(3), pp. 244-263.

Examples

x <- c(541, 1463, 2445, 3438, 4437, 5401, 6392, 8304, 11904, 22261)
atkinson(x)

Atkinson Index for a Generalised Beta Distribution of Second Kind

Description

This function computes the Atkinson index (I(epsilon)) for Generalised Beta Distribution of Second Kind. The function is exact for the values epsilon=0, epsilon=1 and epsilon=2. For other values of epsilon, the function provides a numerical approximation.

Usage

atkinson.GB2(b, a, p, q, epsilon = NULL, ylim = c(0, 1e+06), zeroapprox = 0.01)

Arguments

b

the parameter b of the Dagum distribution as defined by Kleiber and Kotz (2003).

a

the parameter a of the Dagum distribution as defined by Kleiber and Kotz (2003).

p

the parameter p of the Dagum distribution as defined by Kleiber and Kotz (2003).

q

the parameter q of the Dagum distribution as defined by Kleiber and Kotz (2003).

epsilon

inequality aversion parameter as denoted by Atkinson (1970). The default is epsilon=1.

ylim

limits of the interval of y considered needed for the approximation of the entropy measure. The default is [0,1e+06].

zeroapprox

an approximation for zero needed for the approximation of the entropy measure. The default is 0.01.

Value

returns the selected Atkinson inequality index.

Author(s)

Alexander Sohn

References

Atkinson, A.B. (1970): On the Measurment of Inequality, in: Journal of Economic Theory, Vol. 2(3), pp. 244-263.

Cowell, F.A. (2000): Measurement of Inequality, in: Atkinson and Bourguignon (eds.), Handbook of Income Distribution, pp. 87-166, Elsevier, Amsterdam.

Examples

a.test<- 4
b.test<- 20000
p.test<- 0.7
q.test<- 1
epsilon.test<-1
GB2sample<-rGB2(1000,b.test,a.test,p.test,q.test)
atkinson.GB2(b.test,a.test,p.test,q.test,epsilon=epsilon.test,ylim=c(0,1e+07))
atkinson(GB2sample, epsilon.test)

Atkinson Index for an Income Distribution

Description

This function approximates the Atkinson index for a distribution specified by a vector of densities and a corresponding income vector. A point mass at zero is allowed.

Usage

atkinson.den(incs, dens, epsilon = 1, pm0 = NA, 
lower = NULL, upper = NULL, zero.approx = NULL)

Arguments

incs

a vector with income values.

dens

a vector with the corresponding densities.

epsilon

inequality aversion parameter as denoted by Atkinson (1970). The default is epsilon=1.

pm0

the point mass for zero incomes. If not specified no point mass is assumed.

lower

the lower bound of the income range considered.

upper

the upper bound of the income range considered.

zero.approx

a scalar which replaces zero-incomes, such that the Atkinson index involving a logarithm return finite values.

Value

AIM

the approximation of the selected Atkinson inequality measure.

epsilon

the inequality aversion parameter used.

mean

the approximated expected value of the distribution.

pm0

the point mass for zero incomes used.

lower

the lower bound of the income range considered used.

upper

the upper bound of the income range considered used.

zero.approx

the zero approximation used.

Author(s)

Alexander Sohn

References

Atkinson, A.B. (1970): On the Measurment of Inequality, in: Journal of Economic Theory, Vol. 2(3), pp. 244-263.

Examples

## without point mass at zero
incs<-seq(0,500,by=0.01)
dens<-dLOGNO(incs,2,1)
plot(incs,dens,type="l",xlim=c(0,100))
atkinson.den(incs=incs,dens=dens,epsilon=1)$AIM
atkinson(rLOGNO(50000,2,1),epsilon=1)
atkinson.den(incs=incs,dens=dens,epsilon=0.5)$AIM
atkinson(rLOGNO(50000,2,1),epsilon=0.5)

## with point mass at zero
incs<-c(seq(0,100,by=0.1),seq(100.1,1000,by=1),seq(1001,10000,by=10))
dens<-dLOGNO(incs,2,1)/2
dens[1]<-0.5
plot(incs,dens,type="l",ylim=c(0,max(dens[-1])),xlim=c(0,100))
#without zero approx zeros
atkinson.den(incs=incs,dens=dens,epsilon=1,pm0=0.5)$AIM
atkinson(c(rep(0,25000),rLOGNO(25000,2,1)),epsilon=1)
atkinson.den(incs=incs,dens=dens,epsilon=0.5,pm0=0.5)$AIM
atkinson(c(rep(0,25000),rLOGNO(25000,2,1)),epsilon=0.5)
#with zero approximation 
atkinson.den(incs=incs,dens=dens,epsilon=0.5,pm0=0.5,zero.approx=1)$AIM
atkinson(c(rep(1,25000),rLOGNO(25000,2,1)),epsilon=0.5)
atkinson.den(incs=incs,dens=dens,epsilon=1,pm0=0.5,zero.approx=0.01)$AIM
atkinson(c(rep(0.01,250000),rLOGNO(250000,2,1)),epsilon=1)

Atkinson Index for a Mixture of Income Distributions

Description

This function uses Monte Carlo methods to estimate the Atkinson index for a mixture of two continuous income distributions and a point mass for zero-incomes.

Usage

atkinson.md(n, epsilon = 1, dist1, dist2, theta, p0, p1, p2, 
dist.para.table, zero.approx)

Arguments

n

sample size used to estimate the Atkinson index.

epsilon

inequality aversion parameter as denoted by Atkinson (1970). The default is epsilon=1.

dist1

character string with the name of the first continuous distribution used. Must be listed in dist.para.table. Must be equivalent to the respective function of that distribution, e.g. norm for the normal distribution.

dist2

character string with the name of the second continuous distribution used. Must be listed in dist.para.table. Must be equivalent to the respective function of that distribution, e.g. norm for the normal distribution.

theta

vector with the parameters of dist1 and dist2. Order must be the same as in the functions for the distributions.

p0

scalar with probability mass for the point mass.

p1

scalar with probability mass for dist1.

p2

scalar with probability mass for dist2.

dist.para.table

a table of the same form as dist.para.t with distribution name, function name and number of parameters.

zero.approx

a scalar which replaces zero-incomes, such that the Atkinson index involving a logarithm return finite values.

Value

AIM

the selected Atkinson inequality measure.

epsilon

the inequality aversion parameter used.

y

a vector with the simulated incomes to estimate the entropy measure.

y2

a vector with the zero-replaced simulated incomes to estimate the entropy measure.

zero.replace

a logical vector indicating whether a zero has been replaced.

stat

a vector with the simulated group the observation was chosen from. 0 is the point mass, 1 dist1 and 2 dist2.

Author(s)

Alexander Sohn

References

Atkinson, A.B. (1970): On the Measurment of Inequality, in: Journal of Economic Theory, Vol. 2(3), pp. 244-263.

Examples

theta<-c(2,1,5,2)
x<- c(rgamma(50000,2,1),rgamma(50000,5,2))
para<-1

data(dist.para.t)
atkinson.md(10000,para,"gamma","gamma",theta,0,0.5,0.5,dist.para.t,zero.approx=1)$AIM
atkinson(x,1)

Cumulative Density Function of Log-Normal Mixture Distribution

Description

This function yields the cdf of a mixture distribution consisting of a point mass (at the lower end), a uniform distribution (above the point mass and below the log-normal distribution) and a log-normal distribution.

Usage

cdf.mix.LN(q, pi0, thres0 = 0, pi1, thres1, mu, sigma)

Arguments

q

a vector of quantiles.

pi0

the probability mass at thres0.

thres0

the location of the probability mass at the lower end of the distribution.

pi1

the probability mass of the uniform distribution.

thres1

the upper bound of the uniform distribution.

mu

the parameter mu of the Dagum distribution as defined by the function GB2.

sigma

the parameter sigma of the Dagum distribution as defined by the function GB2.

Value

returns the cumulative density for the given quantiles.

Author(s)

Alexander Sohn

References

Sohn, A., Klein, N., Kneib. T. (2014): A New Semiparametric Approach to Analysing Conditional Income Distributions, in: SOEPpapers, No. 676.

Examples

pi0.s<-0.2
pi1.s<-0.1
thres0.s<-0
thres1.s<-25000
mu.s<-10
sigma.s<-2

cdf.mix.LN(50000,pi0.s,thres0.s,pi1.s,thres1.s,mu.s,sigma.s)

Cumulative Density Function of Dagum Mixture Distribution

Description

This function yields the cdf of a mixture distribution consisting of a point mass (at the lower end), a uniform distribution (above the point mass and below the Dagum distribution) and a Dagum distribution.

Usage

cdf.mix.dag(q, pi0, thres0 = 0, pi1, thres1, mu, sigma, nu, tau)

Arguments

q

a vector of quantiles.

pi0

the probability mass at thres0.

thres0

the location of the probability mass at the lower end of the distribution.

pi1

the probability mass of the uniform distribution.

thres1

the upper bound of the uniform distribution.

mu

the parameter mu of the Dagum distribution as defined by the function GB2.

sigma

the parameter sigma of the Dagum distribution as defined by the function GB2.

nu

the parameter nu of the Dagum distribution as defined by the function GB2.

tau

the parameter tau of the Dagum distribution as defined by the function GB2.

Value

returns the cumulative density for the given quantiles.

Author(s)

Alexander Sohn

References

Sohn, A., Klein, N. and Kneib. T. (2014): A New Semiparametric Approach to Analysing Conditional Income Distributions, in: SOEPpapers, No. 676.

Examples

pi0.s<-0.2
pi1.s<-0.1
thres0.s<-0
thres1.s<-25000
mu.s<-20000
sigma.s<-5
nu.s<-0.5
tau.s<-1

cdf.mix.dag(50000,pi0.s,thres0.s,pi1.s,thres1.s,mu.s,sigma.s,nu.s,tau.s)

Coefficient of Variation

Description

This function computes the Coefficient of Variation for a vector of observations.

Usage

coeffvar(x)

Arguments

x

a vector of observations.

Value

cv

returns the coefficient of variation without bias correction.

bccv

returns the coefficient of variation with bias correction.

Warning

Weighting is not properly accounted for in the sample adjustment of bccv!

Author(s)

Alexander Sohn

References

Atkinson, A.B. and Bourguignon, F. (2000): Income Distribution and Economics, in: Atkinson and Bourguignon (eds.), Handbook of Income Distribution, pp. 1-86, Elsevier, Amsterdam.

Examples

# generate vector (of incomes)
x <- c(541, 1463, 2445, 3438, 4437, 5401, 6392, 8304, 11904, 22261)
coeffvar(x)

Simultaneous Confidence Bands

Description

This function computes simultaneous confidence bands for samples of the presumed distribution of the parameter estimator.

Usage

confband.kneib(samples, level = 0.95)

Arguments

samples

matrix containing samples of the presumed distribution of the parameter estimator.

level

the desired confidence level.

Value

lower

a vector containing the lower bound of the confidence band.

upper

a vector containing the lower bound of the confidence band.

Note

This function is taken from the work of T. Krivobokova, T. Kneib and G. Claeskens.

Author(s)

Alexander Sohn

References

T. Krivobokova, T. Kneib, G. Claeskens (2010): Simultaneous Confidence Bands for Penalized Spline Estimators, in: Journal of the American Statistical Association, Vol. 105(490), pp.852-863.

Examples

mu<-1:20
n<-1000
mcmc<-matrix(NA,n,20)
for(i in 1:20){
  mcmc[,i]<- rnorm(n,mu[i],sqrt(i))
}

plot(mu,type="l",ylim=c(-10,30),lwd=3)
lines(confband.pw(mcmc)$lower,lty=2)
lines(confband.pw(mcmc)$upper,lty=2)
lines(confband.kneib(mcmc)$lower,lty=3)
lines(confband.kneib(mcmc)$upper,lty=3)

Pointwise Confidence Bands

Description

This function computes pointwise confidence bands for samples of the presumed distribution of the parameter estimator.

Usage

confband.pw(samples, level = 0.95)

Arguments

samples

matrix containing samples of the presumed distribution of the parameter estimator.

level

the desired confidence level.

Value

lower

a vector containing the lower bound of the confidence band.

upper

a vector containing the lower bound of the confidence band.

Note

This function is mainly derived from the work of T. Krivobokova, T. Kneib and G. Claeskens.

Author(s)

Alexander Sohn

References

T. Krivobokova, T. Kneib, G. Claeskens (2010): Simultaneous Confidence Bands for Penalized Spline Estimators, in: Journal of the American Statistical Association, Vol. 105(490), pp.852-863.

Examples

mu<-1:20
n<-1000
mcmc<-matrix(NA,n,20)
for(i in 1:20){
  mcmc[,i]<- rnorm(n,mu[i],sqrt(i))
}

plot(mu,type="l",ylim=c(-10,30),lwd=3)
lines(confband.pw(mcmc)$lower,lty=2)
lines(confband.pw(mcmc)$upper,lty=2)
lines(confband.kneib(mcmc)$lower,lty=3)
lines(confband.kneib(mcmc)$upper,lty=3)

ACID Simulated Data

Description

This is some simulated income data from a mixture model as used in Sohn et al (2014).

Usage

data(dat)

Format

The format is: List of 4 $ dag.para:'data.frame': 8 obs. of 1 variable: ..$ parameters: num [1:8] 0.2 0.1 0 25000 20000 5 0.5 1 $ dag.s :'data.frame': 100 obs. of 3 variables: ..$ cat: int [1:100] 3 1 3 1 2 3 3 1 3 3 ... ..$ y : num [1:100] 36410 0 58165 0 15034 ... ..$ w : int [1:100] 1 1 1 2 1 3 2 1 1 1 ... $ LN.para :'data.frame': 6 obs. of 1 variable: ..$ parameters: num [1:6] 0.2 0.1 0 25000 10 2 $ LN.s :'data.frame': 100 obs. of 3 variables: ..$ cat: int [1:100] 3 3 1 3 3 3 3 3 3 3 ... ..$ y : num [1:100] 29614 29549 0 33068 463941 ... ..$ w : int [1:100] 1 2 1 1 1 1 1 2 1 1 ...

Details

The data contains information on whether the person is unemployed (cat=1), precariously employed (cat=2) or in standard employment(cat=3), the corresponding parameters used to generate the truncated distribution - both for Log-normal and Dagum.

References

Sohn, A., Klein, N., Kneib. T. (2014): A New Semiparametric Approach to Analysing Conditional Income Distributions, in: SOEPpapers, No. 676.

Examples

data(dat)
str(dat)

Density for a Mixture of Income Distributions

Description

This function computes the p-value for a mixture of two continuous income distributions and a point mass for zero-incomes.

Usage

den.md(y, dist1, dist2, theta, p0, p1, p2, dist.para.table)

Arguments

y

a vector with incomes. If a zero income is included, it must be the first element.

dist1

dist2

theta

vector with the parameters of dist1 and dist2. Order must be the same as in the functions for the distributions.

p0

scalar with probability mass for the point mass.

p1

scalar with probability mass for dist1.

p2

scalar with probability mass for dist2.

dist.para.table

a table of the same form as dist.para.t with distribution name, function name and number of parameters.

Value

returns the density for given values of y.

Author(s)

Alexander Sohn

Examples

data(dist.para.t)
ygrid<-seq(0,20,by=0.1)#c(seq(0,1e5,by=100),seq(1.1e5,1e6,by=100000))
theta<-c(5,1,10,1.5)
p0<-0.2
p1<-0.3
p2<-0.5
n <-100000
y.sim <- ysample.md(n, "norm", "norm", theta, p0, p1, p2, dist.para.t)
den<-den.md(ygrid,"norm", "norm", theta, 
              p0, p1, p2, dist.para.table=dist.para.t)
hist(y.sim,freq=FALSE)
#hist(y.sim,breaks=c(seq(0,1e5,by=100),seq(1.1e5,1e6,by=100000)),xlim=c(0,2e4),ylim=c(0,0.001))
lines(ygrid,den,col=2)

Distributions and their Parameters

Description

A data frame providing information on the number of parameters of distributions used for analysing conditional income distributions.

Usage

data(dist.para.t)

Format

A data frame with the following 3 variables.

Distribution: name of the distribution.
dist: function of the distribution.
Parameters: the number of parameters for the distribution.

Examples

data(dist.para.t)
dist.para.t

Measures of the Generalised Entropy Family

Description

This function computes the Measures of the Generalised Entropy Family for a vector of observations.

Usage

entropy(x, alpha = 1)

Arguments

x

a vector of observations.

alpha

the parameter for the generalised entropy family of measures, denoted by alpha by Cowell (2000). Note that this parameter notation differs from the notation used in the ineq package.

Value

returns the entropy measure.

Author(s)

Alexander Sohn

References

Cowell, F.A. (2000): Measurement of Inequality, in: Atkinson and Bourguignon (eds.), Handbook of Income Distribution, pp. 1-86, Elsevier, Amsterdam.

Examples

# generate vector (of incomes)
x <- c(541, 1463, 2445, 3438, 4437, 5401, 6392, 8304, 11904, 22261)
entropy(x)

Entropy Measures for a Generalised Beta Distribution of Second Kind

Description

This function computes four standard entropy measures from the generalised entropy class of inequality indices (I(alpha)) for Generalised Beta Distribution of Second Kind, namely the mean logarithmic deviation (I(0)), the Theil index (I(1)) as well as a bottom-sensitive index (I(-1)) and a top-sensitive index (I(2)). For other values of alpha, the function provides a numerical approximation.

Usage

entropy.GB2(b, a, p, q, alpha = NULL, ylim = c(0, 1e+06), zeroapprox = 0.01)

Arguments

b

the parameter b of the Dagum distribution as defined by Kleiber and Kotz (2003).

a

the parameter a of the Dagum distribution as defined by Kleiber and Kotz (2003).

p

the parameter p of the Dagum distribution as defined by Kleiber and Kotz (2003).

q

the parameter q of the Dagum distribution as defined by Kleiber and Kotz (2003).

alpha

measure for the entropy measure as denoted by Cowell (2000). The default is alpha=1, i.e. the Theil Index.

ylim

limits of the interval of y considered needed for the approximation of the entropy measure. The default is [0,1e+06].

zeroapprox

an approximation for zero needed for the approximation of the entropy measure. The default is 0.01.

Value

returns the selected entropy measure.

Author(s)

Alexander Sohn

References

Kleiber, C. and Kotz, S. (2003): Statistical Size Distributions in Economics and Actuarial Sciences, Wiley, Hoboken.

Cowell, F.A. (2000): Measurement of Inequality, in: Atkinson and Bourguignon (eds.), Handbook of Income Distribution, pp. 87-166, Elsevier, Amsterdam.

Jenkins, S.P. (2009): Distributionally-Sensitive Inequality Indices and the GB2 Income Distribution, in: Review of Income and Wealth, Vol. 55(2), pp.392-398.

Examples

a.test<- 4
b.test<- 20000
p.test<- 0.7
q.test<- 1
alpha.test<-1
GB2sample<-rGB2(1000,b.test,a.test,p.test,q.test)
entropy.GB2(b.test,a.test,p.test,q.test,alpha=alpha.test,ylim=c(0,1e+07))
entropy(GB2sample, alpha.test)

Generalised Entropy Measure for a Mixture of Income Distributions

Description

This function uses Monte Carlo methods to estimate an entropy measure for a mixture of two continuous income distributions and a point mass for zero-incomes.

Usage

entropy.md(n, alpha = 1, dist1, dist2, theta, 
p0, p1, p2, dist.para.table, zero.approx)

Arguments

n

sample size used to estimate the entropy measure.

alpha

the parameter for the generalised entropy family of measures, denoted by alpha by Cowell (2000). Note that this parameter notation differs from the notation used in the ineq package.

dist1

dist2

theta

vector with the parameters of dist1 and dist2. Order must be the same as in the functions for the distributions.

p0

scalar with probability mass for the point mass.

p1

scalar with probability mass for dist1.

p2

scalar with probability mass for dist2.

dist.para.table

a table of the same form as dist.para.t with distribution name, function name and number of parameters.

zero.approx

a scalar which replaces zero-incomes (and negative incomes), such that entropy measures involving a logarithm return finite values.

Value

entropy

the estimated entropy measure.

alpha

the entropy parameter used.

y

a vector with the simulated incomes to estimate the entropy measure.

y2

a vector with the zero-replaced simulated incomes to estimate the entropy measure.

zero.replace

a logical vector indicating whether a zero has been replaced.

stat

a vector with the simulated group the observation was chosen from. 0 is the point mass, 1 dist1 and 2 dist2.

Author(s)

Alexander Sohn

References

Cowell, F.A. (2000): Measurement of Inequality, in: Atkinson and Bourguignon (eds.), Handbook of Income Distribution, pp. 87-166, Elsevier, Amsterdam.

Examples

 
theta<-c(2,1,5,2)
x<- c(rgamma(500,2,1),rgamma(500,5,2))
para<-1
entropy(x,para)
data(dist.para.t)
entropy.md(100,para,"gamma","gamma",theta,0,0.5,0.5,dist.para.t,zero.approx=1)$entropy

Fractional Ranks

Description

This function computes fractional ranks which are required for the S-Gini coefficient.

Usage

frac.ranks(x, w = NULL)

Arguments

x

a vector with sorted income values.

w

a vector of weights.

Value

returns the fractional ranks.

Author(s)

Alexander Sohn

References

van Kerm, P. (2009): sgini - Generalized Gini and Concentration coefficients (with factor decomposition) in Stata', CEPS/INSTEAD, Differdange, Luxembourg.

Gini Coefficient

Description

This function computes the Gini coefficient for a vector of observations.

Usage

gini(x)

Arguments

x

a vector of observations.

Value

Gini

the Gini coefficient for the sample.

bcGini

the bias-corrected Gini coefficient for the sample.

Author(s)

Alexander Sohn

References

Cowell, F.A. (2000): Measurement of Inequality, in: Atkinson and Bourguignon (eds.), Handbook of Income Distribution, pp. 87-166, Elsevier, Amsterdam.

Examples

# generate vector (of incomes)
x <- c(541, 1463, 2445, 3438, 4437, 5401, 6392, 8304, 11904, 22261)
gini(x)

Gini Coefficient for the Dagum Distribution

Description

This function computes the Gini coefficient for the Dagum Distribution.

Usage

gini.Dag(a, p)

Arguments

a

the parameter a of the Dagum distribution as defined by Kleiber and Kotz (2003).

p

the parameter p of the Dagum distribution as defined by Kleiber and Kotz (2003).

Value

returns the Gini coefficient.

Author(s)

Alexander Sohn

References

Cowell, F.A. (2000): Measurement of Inequality, in: Atkinson and Bourguignon (eds.), Handbook of Income Distribution, pp. 1-86, Elsevier, Amsterdam.

Examples

a.test<- 4
b.test<- 20000
p.test<- 0.7
alpha.test<-1
GB2sample<-rGB2(10000,b.test,a.test,p.test,1)
gini.Dag(a.test,p.test)
gini(GB2sample)

Gini Coefficient for an Income Distribution

Description

This function approximates the Gini coefficient for a distribution specified by a vector of densities and a corresponding income vector. A point mass at zero is allowed.

Usage

gini.den(incs, dens, pm0 = NA, 
lower = NULL, upper = NULL)

Arguments

incs

a vector with sorted income values.

dens

a vector with the corresponding densities.

pm0

the point mass for zero incomes. If not specified no point mass is assumed.

lower

the lower bound of the income range considered.

upper

the upper bound of the income range considered.

Value

Gini

the approximation of the Gini coefficient.

pm0

the point mass for zero incomes used.

lower

the lower bound of the income range considered used.

upper

the upper bound of the income range considered used.

Author(s)

Alexander Sohn

References

Cowell, F.A. (2000): Measurement of Inequality, in: Atkinson and Bourguignon (eds.), Handbook of Income Distribution, pp. 1-86, Elsevier, Amsterdam.

Examples

mu<-2
sigma<-1
incs<-c(seq(0,500,by=0.01),seq(501,50000,by=1))
dens<-dLOGNO(incs,mu,sigma)
plot(incs,dens,type="l",xlim=c(0,100))
gini.den(incs=incs,dens=dens)$Gini
gini(rLOGNO(5000000,mu,sigma))$Gini
2*pnorm(sigma/sqrt(2))-1 #theoretical Gini

Gini Coefficient for the Gamma Distribution

Description

This function computes the Gini coefficient for the gamma distribution.

Usage

gini.gamma(p)

Arguments

p

the shape parameter p of the gamma distribution as defined by Kleiber and Kotz (2003).

Value

returns the Gini coefficient.

Author(s)

Alexander Sohn

References

Cowell, F.A. (2000): Measurement of Inequality, in: Atkinson and Bourguignon (eds.), Handbook of Income Distribution, pp. 1-86, Elsevier, Amsterdam.

Kleiber, C. and Kotz, S. (2003): Statistical Size Distributions in Economics and Actuarial Sciences, Wiley, Hoboken.

Examples

shape.test <- 5
scale.test  <- 50000
y <- rgamma(10000,shape=shape.test,scale=scale.test)
gini(y)
gini.gamma(shape.test)

Gini Coefficient for a Mixture of Income Distributions

Description

This function uses Monte Carlo methods to estimate the Gini coefficient for a mixture of two continuous income distributions and a point mass for zero-incomes.

Usage

gini.md(n, dist1, dist2, theta, 
p0, p1, p2, dist.para.table)

Arguments

n

sample size used to estimate the gini coefficient.

dist1

dist2

theta

vector with the parameters of dist1 and dist2. Order must be the same as in the functions for the distributions.

p0

scalar with probability mass for the point mass.

p1

scalar with probability mass for dist1.

p2

scalar with probability mass for dist2.

dist.para.table

a table of the same form as dist.para.t with distribution name, function name and number of parameters.

Value

gini

the estimated Gini coefficient.

y

a vector with the simulated incomes to estimate the Gini coefficient.

stat

a vector with the simulated group the observation was chosen from. 0 is the point mass, 1 dist1 and 2 dist2.

Author(s)

Alexander Sohn

References

Cowell, F.A. (2000): Measurement of Inequality, in: Atkinson and Bourguignon (eds.), Handbook of Income Distribution, pp. 87-166, Elsevier, Amsterdam.

Examples

 
theta<-c(2,1,5,2)
x<- c(rnorm(500,2,1),rnorm(500,5,2))
gini(x)$Gini
data(dist.para.t)
gini.md(1000,"norm","norm",theta,0,0.5,0.5,dist.para.t)$gini

Three Inequality Measures for a Mixture of Income Distributions

Description

This function uses Monte Carlo methods to estimate an the mean logarithmic deviation, the Theil Index and the Gini Coefficient for a mixture of two continuous income distributions and a point mass for zero-incomes.

Usage

ineq.md(n, dist1, dist2, theta, 
p0, p1, p2, dist.para.table, zero.approx)

Arguments

n

sample size used to estimate the gini coefficient.

dist1

dist2

theta

vector with the parameters of dist1 and dist2. Order must be the same as in the functions for the distributions.

p0

scalar with probability mass for the point mass.

p1

scalar with probability mass for dist1.

p2

scalar with probability mass for dist2.

dist.para.table

a table of the same form as dist.para.t with distribution name, function name and number of parameters.

zero.approx

a scalar which replaces zero-incomes (and negative incomes), such that entropy measures involving a logarithm return finite values.

Value

MLD

the estimated mean logarithmic deviation.

Theil

the estimated Theil index.

Gini

the estimated Gini coefficient.

y

a vector with the simulated incomes to estimate the entropy measure.

y2

a vector with the zero-replaced simulated incomes to estimate the entropy measure.

zero.replace

a logical vector indicating whether a zero has been replaced.

stat

a vector with the simulated group the observation was chosen from. 0 is the point mass, 1 dist1 and 2 dist2.

Author(s)

Alexander Sohn

References

Cowell, F.A. (2000): Measurement of Inequality, in: Atkinson and Bourguignon (eds.), Handbook of Income Distribution, pp. 87-166, Elsevier, Amsterdam.

Examples

 
theta<-c(0,1,5,2)
x<- c(rgamma(500,2,1),rgamma(500,5,2))
entropy(x,0)
entropy(x,1)
gini(x)$Gini
data(dist.para.t)
im<-ineq.md(100,"gamma","gamma",theta,0,0.5,0.5,dist.para.t,zero.approx=1)
im$MLD
im$Theil
im$Gini

k-th Moment of the Generalised Beta Distribution of Second Kind

Description

Calculates the k-th moment of the Generalised Beta Distribution of Second Kind.

Usage

km.GB2(b, a, p, q, k)

Arguments

b

the parameter b of the Dagum distribution as defined by Kleiber and Kotz (2003).

a

the parameter a of the Dagum distribution as defined by Kleiber and Kotz (2003).

p

the parameter p of the Dagum distribution as defined by Kleiber and Kotz (2003).

q

the parameter q of the Dagum distribution as defined by Kleiber and Kotz (2003).

k

order of the moment desired.

Value

returns the k-th moment.

Author(s)

Alexander Sohn

References

Kleiber, C. and Kotz, S. (2003): Statistical Size Distributions in Economics and Actuarial Sciences, Wiley, Hoboken.

Examples

a.test<- 4
b.test<- 20000
p.test<- 0.7
q.test<- 1
alpha.test<-1
GB2sample<-rGB2(10000,b.test,a.test,p.test,q.test)
km.GB2(b.test,a.test,p.test,q.test,k=1)
mean(GB2sample)

Plot Comparing Parametric and Empirical Cumulative Density Functions

Description

This function plots a graph entailing the empirical cdf and the parametrically specified cdf composed of a mixture distribution either by cdf.mix.dag or cdf.mix.LN.

Usage

midks.plot(x.seq, y, dist, w.emp = NULL, ...)

Arguments

x.seq

the sequence on the x-axis for which the parametric distribution is plotted.

y

a vector of observed incomes.

dist

a function specifying the parametric cdf.

w.emp

the weights of the observations contained in y.

...

arguments to be passed to dist.

Author(s)

Alexander sohn

Examples

# parameter values
pi0.s<-0.2
pi1.s<-0.1
thres0.s<-0
thres1.s<-25000
mu.s<-20000
sigma.s<-5
nu.s<-0.5
tau.s<-1
x.seq<-seq(0,200000,by=1000)

# generate sample
n<-100
s<-as.data.frame(matrix(NA,n,3))
names(s)<-c("cat","y","w")
s[,1]<-sample(1:3,n,replace=TRUE,prob=c(pi0.s,pi1.s,1-pi0.s-pi1.s))
s[,3]<-rep(1,n)
for(i in 1:n){
  if(s$cat[i]==1){s$y[i]<-0
  }else if(s$cat[i]==2){s$y[i]<-runif(1,thres0.s,thres1.s)
  }else s$y[i]<-rGB2(1,mu=mu.s,sigma=sigma.s,nu=nu.s,tau=tau.s)+thres1.s
}
# display
midks.plot(x.seq,s$y,dist=cdf.mix.dag,pi0=pi0.s,thres0=thres0.s,pi1=pi1.s,
thres1=thres1.s,mu=mu.s,sigma=sigma.s,nu=nu.s,tau=tau.s)

Kolmogorov-Smirnov Test assessing a Parametric Mixture for a Conditional Income Distribution

Description

This function performs a Kolmogorov-Smirnov test for a parametrically specified cdf composed of a mixture distribution either by cdf.mix.dag or cdf.mix.LN.

Usage

midks.test(x, y, ..., w = NULL, pmt = NULL)

Arguments

x

a vector of observed incomes.

y

a function specifying the parametric cdf.

...

arguments to be passed to y.

w

the weights of the observations contained in y.

pmt

point mass threshold equivalent to thres0 in y.

Value

statistic

returns the test statistic.

method

returns the methodology - currently always One-sample KS-test.

diffpm

the difference of the probability for the point mass.

diff1

the upper difference between for the continuous part of the cdfs.

diff2

the lower difference between for the continuous part of the cdfs.

Author(s)

Alexander Sohn

References

Sohn, A., Klein, N. and Kneib. T. (2014): A New Semiparametric Approach to Analysing Conditional Income Distributions, in: SOEPpapers, No. 676.

Examples

# parameter values
pi0.s<-0.2
pi1.s<-0.1
thres0.s<-0
thres1.s<-25000
mu.s<-20000
sigma.s<-5
nu.s<-0.5
tau.s<-1

# generate sample
n<-100
s<-as.data.frame(matrix(NA,n,3))
names(s)<-c("cat","y","w")
s[,1]<-sample(1:3,n,replace=TRUE,prob=c(pi0.s,pi1.s,1-pi0.s-pi1.s))
s[,3]<-rep(1,n)
for(i in 1:n){
  if(s$cat[i]==1){s$y[i]<-0
  }else if(s$cat[i]==2){s$y[i]<-runif(1,thres0.s,thres1.s)
  }else s$y[i]<-rGB2(1,mu=mu.s,sigma=sigma.s,nu=nu.s,tau=tau.s)+thres1.s
}

# midks.test
midks.test(s$y,cdf.mix.dag,pi0=pi0.s,thres0=thres0.s,pi1=pi1.s,thres1=thres1.s,mu=mu.s,
sigma=sigma.s,nu=nu.s,tau=tau.s,w=s$w,pmt=thres0.s)$statistic

Parameter estimators obtained from Structured Additive Distributional Regression

Description

A list containing parameter estimates as obtained from Structured Additive Distributional Regression

Usage

data(params)

Format

The format is: List of 16 $ mcmcsize : num 1000 $ ages : int [1:40] 21 22 23 24 25 26 27 28 29 30 ... $ unems : num [1:23] 0 1 2 3 4 5 6 7 8 9 ... $ educlvls : num [1:2] -1 1 $ bulas : chr [1:16] "SH" "HH" "NDS" "Bremen" ... $ aft.v : num [1:3447] 4.85 6.5 5.92 5.76 6.05 ... $ bft.v : num [1:3447] 78169 65520 47184 58763 46188 ... $ cft.v : num [1:3447] 1.177 0.299 0.818 0.522 0.836 ... $ mupt.v : num [1:3447] 10.21 9.46 9.66 9.77 9.68 ... $ sigmapt.v: num [1:3447] 1.07 1.25 1.85 1.21 1.74 ... $ muemp.v : num [1:3447] 3.25 2.68 2.08 3.53 2.43 ... $ muunemp.v: num [1:3447] -2.691 -0.813 -1.919 -1.542 -1.765 ... $ punemp.v : num [1:3447] 0.0658 0.3104 0.1314 0.18 0.1496 ... $ pemp.v : num [1:3447] 0.934 0.69 0.869 0.82 0.85 ... $ pft.v : num [1:3447] 0.898 0.644 0.77 0.796 0.78 ... $ ppt.v : num [1:3447] 0.0359 0.0452 0.0987 0.024 0.0706 ...

Examples

data(params)
str(params)
## maybe str(params) ; plot(params) ...

Pen's Parade

Description

This function plots Pen's parade.

Usage

pens.parade(x, bodies = TRUE, feet = 0, ...)

Arguments

x

a vector of observed incomes.

bodies

a logical value indicating whether lines, i.e. the bodies, should be drawn.

feet

a numeric value indicating where the lines originate.

...

additional arguments passed to the plot function.

Author(s)

Alexander Sohn

References

Atkinson, A.B. (1975): The Economics of Inequality, Cleardon Press, Oxford.

Examples

a.test<- 4
b.test<- 20000
p.test<- 0.7
q.test<- 1
alpha.test<-1
GB2sample<-rGB2(100,b.test,a.test,p.test,q.test)
pens.parade( GB2sample)

Polarisation Measure from Esteban, Gradin and Ray (2007)

Description

This function computes the polarisation measure proposed in Esteban, Gradin and Ray (2007) which accounts for deviations from an n-spike representation of strata in society.

Usage

polarisation.EGR(alpha, beta, rho, y, f = NULL, dist = NULL, 
weights = NULL, pm0 = NA, lower = NULL, upper = NULL, ...)

Arguments

alpha

a scalar containing the alpha parameter from Esteban and Ray (1994) on the sensitivity to polarisation.

beta

a scalar containing the beta parameter from Esteban, Gradin and Ray (2007) on the weight assigned to the error in the n-spike representation.

rho

a dataframe with the group means in the first column and their respective population shares in the second. The groups need to be exogenously defined.

Note: the two columns should be named means and shares respectively. Otherwise a warning will appear.

y

a vector of incomes. If f is NULL and dist is NULL, this includes all incomes of all observations in the sample, i.e. all observations comprising the aggregate distribution. If either f or dist is not NULL, then this gives the incomes where the density is evaluated.

f

a vector of user-defined densities of the aggregate distribution for the given incomes in y.

dist

character string with the name of the distribution used. Must be equivalent to the respective function of that distribution, e.g. norm for the normal distribution.

weights

an optional vector of weights to be used in the fitting process. Should be NULL or a numeric vector. If non-NULL, observations in y are weighted accordingly.

pm0

the point mass for zero incomes used in the gini.den function. If not specified no point mass is assumed.

lower

the lower bound of the income range considered used in the gini.den function.

upper

the upper bound of the income range considered used in the gini.den function.

...

arguments to be passed to the distribution function used, e.g. mean and sd for the normal distribution.

Value

P

the polarisation measure proposed by Esteban, Gradin and Ray (2007).

PG

the adjusted polarisation measure proposed by Gradin (2000).

alpha

the alpha parameter used.

beta

the beta parameter used.

beta

the distribution option used, i.e. whether only y, f or dist was used.

Author(s)

Alexander Sohn

References

Esteban, J. and Ray, D. (1994): On the Measurment of Polarization, in: Econometrica, Vol. 62(4), pp. 819-851.

Esteban, J., Gradin, C. and Ray, D. (2007): Extensions of a Measure of Polarization, with an Application to the Income Distribution of five OECD Countries.

Gradin, C. (2000): Polarization by Sub-populations in Spain, 1973-91, in Review of Income and Wealth, Vol. 46(4), pp.457-474.

Examples

## example 1
y<-rnorm(1000,5,0.5)
y<-sort(y)
m.y<-mean(y)
sd.y<-sd(y)
y1<-y[1:(length(y)/4)]
m.y1<-mean(y1)
sd.y1<-sd(y1)
y2<-y[(length(y)/4+1):length(y)]
m.y2<-mean(y2)
sd.y2<-sd(y2)
means<-c(m.y1,m.y2)
share1<- length(y1)/length(y)
share2<- length(y2)/length(y)
shares<- c(share1,share2)
rho<-data.frame(means=means,shares=shares)
alpha<-1
beta<-1
den<-density(y)
polarisation.ER(alpha,rho,comp=FALSE)
polarisation.EGR(alpha,beta,rho,y)$P
polarisation.EGR(alpha,beta,rho,y=den$x,f=den$y)$P
polarisation.EGR(alpha,beta,rho,y=seq(0,10,by=0.1),dist="norm",
mean=m.y,sd=sd.y)$P
polarisation.EGR(alpha,beta,rho,y=seq(0,10,by=0.1),dist="norm",
mean=m.y,sd=sd.y)$PG

## example 2
y1<-rnorm(100,5,1)
y2<-rnorm(100,1,0.1)
y <- c(y1,y2)
m.y1<-mean(y1)
sd.y1<-sd(y1)
m.y2<-mean(y2)
sd.y2<-sd(y2)
means<-c(m.y1,m.y2)
share1<- length(y1)/length(y)
share2<- length(y2)/length(y)
shares<- c(share1,share2)
rho<-data.frame(means=means,shares=shares)
alpha<-1
beta<-1
polarisation.EGR(alpha,beta,rho,y=seq(0,10,by=0.1),dist="norm",
                 mean=c(m.y1,m.y2),sd=c(sd.y1,sd.y2))$P

Polarisation Measure from Esteban and and Ray (1994)

Description

This function computes the polarisation measure proposed in Esteban and and Ray (1994).

Usage

polarisation.ER(alpha, rho, comp = FALSE)

Arguments

alpha

a scalar containing the alpha parameter from Esteban and Ray (1994) on the sensitivity to polarisation.

rho

a dataframe with the group means in the first column and their respective population shares in the second. The groups need to be exogenously defined.

Note: the two columns should be named means and ”shares” respectively. Otherwise a warning will appear.

comp

logical; if TRUE, all components pf p_i^(a+alpha)*p_j*abs(y_i-y_j)

Value

P

the polarisation measure proposed by Esteban and Ray (1994).

means

the means stored in rho.

shares

the shares stored in rho..

ERcomp

if comp is TRUE, the components aggregated in P.

Author(s)

Alexander Sohn

References

Esteban, J. and Ray, D. (1994): On the Measurment of Polarization, in: Econometrica, Vol. 62(4), pp. 819-851.

Examples

means<-rnorm(10)+5
shares<-  rep(1,length(means))
shares<-shares/sum(shares)
rho<-data.frame(means=means,shares=shares)
alpha<-1
polarisation.ER(alpha,rho,comp=FALSE)

P-Value for a Mixture of Income Distributions

Description

This function computes the p-value for a mixture of two continuous income distributions and a point mass for zero-incomes.

Usage

pval.md(y, dist1, dist2, theta, p0, p1, p2, dist.para.table)

Arguments

y

a vector with incomes. If a zero income is included, it must be the first element.

dist1

dist2

theta

vector with the parameters of dist1 and dist2. Order must be the same as in the functions for the distributions.

p0

scalar with probability mass for the point mass.

p1

scalar with probability mass for dist1.

p2

scalar with probability mass for dist2.

dist.para.table

a table of the same form as dist.para.t with distribution name, function name and number of parameters.

Value

returns the p-value.

Author(s)

Alexander Sohn

Examples

data(dist.para.t)
ygrid<-seq(0,1e5,by=1000)
theta<-c(5,1,10,3)
p0<-0.2
p1<-0.3
p2<-0.5
n <-10000
y.sim <- ysample.md(n, "LOGNO", "LOGNO", theta, p0, p1, p2, dist.para.t)
pval<-pval.md(ygrid,"LOGNO", "LOGNO", theta, 
              p0, p1, p2, dist.para.table=dist.para.t)
mean(y.sim<=ygrid[10])
pval[10]

Misspecification Test assessing a Parametric Conditional Income Distribution

Description

This function performs a misspecificaton test for a parametrically specified cdf estimated by (Bayesian) Structured Additive Distributional Regression.

Usage

sadr.test(data, y.pos = NULL, dist1, dist2, params.m, mcmc = TRUE, mcmc.params.a,
 ygrid, bsrep = 10, n.startvals = 300, dist.para.table)

Arguments

data

a dataframe including dependent variable and all explanatory variables.

y.pos

an integer indicating the position of the dependent variable in the dataframe.

dist1

dist2

params.m

a matrix with the estimated parameter values (in colums) for each individual (in rows). The order of the parameters must be as follows: parameters for the first distribution, parameters for the second distribution, probability of zero income, probability of dist1, probability of dist2 and probability of dist1 given employment/non-zero income.

mcmc

logical; if TRUE, uncertainty as provided by the MCMC samples is considered.

mcmc.params.a

an array, with the mcmc samples for all the parameters specified by structured additive distributional regression. In the first dimension should be the MCMC realisations, in the second dimension the individuals and in the third the parameters. The order of the parameters must be as follows: parameters for the first distribution, parameters for the second distribution, probability of zero income, probability of dist1, probability of dist2 and probability of dist1 given employment/non-zero income.

ygrid

vector yielding the grid on which the cdf is specified.

bsrep

integer giving the number of bootstrap repitions in order to determine the distributions of the test statistics under the null.

n.startvals

integer giving the maximum number of observations used to estimate the test statistic.

dist.para.table

a table of the same form as dist.para.t with distribution name, function name and number of parameters.

Value

teststat.ks

Kolmogorov-Smirnov test statistic.

pval.ks

p-value based on the Kolmogorov-Smirnov test statistic.

teststat.cvm

Cramer-von-Mises test statistic.

pval.cvm

p-value based on the Cramer-von-Mises test statistic.

test

type cdf considered for the test.

param.distributions

parametric distributions assumed for dist1 and dist2.

teststat.ks.bs

bootstrap results of Kolmogorov-Smirnov test statistic under null.

teststat.cvm.bs

bootstrap results of Cramer-von-Mises test statistic under null.

Author(s)

Alexander Sohn

References

Rothe, C. and Wied, D. (2013): Misspecification Testing in a Class of Conditional Distributional Models, in: Journal of the American Statistical Association, Vol. 108(501), pp.314-324.

Sohn, A. (forthcoming): Scars from the Past and Future Earning Distributions.

Examples

# ### functions not run - take considerable time!
# 
# library(acid)
# data(dist.para.t)
# data(params)
# ### example one - two normals, no mcmc
# dist1<-"norm"
# dist2<-"norm"
# ## generating data
# set.seed(1234)
# n<-1000
# sigma<-0.1
# X.theta<-c(1,10,1,10)
# X.gen<-function(n,paras){
#   X<-matrix(c(round(runif(n,paras[1],paras[2])),round(runif(n,paras[3],
#             paras[4]))),ncol=2)
#   return(X)
# }
# X <- X.gen(n,X.theta)
# beta.mu1   <- 1
# beta.sigma1<- 0.1
# beta.mu2   <- 2
# beta.sigma2<- 0.1
# pi0        <- 0.3
# pi01       <- 0.8
# pi1        <- (1-pi0)*pi01
# pi2        <- 1-pi0-pi1
# 
# params.m<-matrix(NA,n,8)
# params.m[,1]<-(0+beta.mu1)*X[,1]
# params.m[,2]<-(0+beta.sigma1)*X[,1]
# params.m[,3]<-(0+beta.mu2)*X[,2]
# params.m[,4]<-(0+beta.sigma2)*X[,2]
# params.m[,5]<-pi0
# params.m[,6]<-pi1
# params.m[,7]<-pi2
# params.m[,8]<-pi01
# 
# params.mF<-matrix(NA,n,8)
# params.mF[,1]<-(10+beta.mu1)*X[,1]
# params.mF[,2]<-(0+beta.sigma1)*X[,1]
# params.mF[,3]<-(0+beta.mu2)*X[,2]
# params.mF[,4]<-(2+beta.sigma2)*X[,2]
# params.mF[,5]<-pi0
# params.mF[,6]<-pi1
# params.mF[,7]<-pi2
# params.mF[,8]<-pi01
# # starting repititions
# reps<-30
# tsreps1T<-rep(NA,reps)
# tsreps2T<-rep(NA,reps)
# tsreps1F<-rep(NA,reps)
# tsreps2F<-rep(NA,reps)
# sys.t<-Sys.time()
# for(r in 1:reps){
#   Y <- rep(NA,n)
#   for(i in 1:n){
#     Y[i] <- ysample.md(1,dist1,dist2,theta=params.m[i,1:4],params.m[i,5],
#     params.m[i,6],params.m[i,7],dist.para.t)
#   }
#   dat<-cbind(Y,X)
#   y.pos<-1
#   ygrid<-seq(min(Y),round(max(Y)*1.2,-1),by=1)  
#   tsT<-sadr.test(data=dat,y.pos=NULL,dist1="norm",dist2="norm",
#   params.m=params.m,mcmc=FALSE,mcmc.params=NA,ygrid=ygrid, bsrep=100,
#   n.startvals=30000,dist.para.table=dist.para.t)
#   tsreps1T[r]<-tsT$pval.ks
#   tsreps2T[r]<-tsT$pval.cvm
#   tsF<-sadr.test(data=dat,y.pos=NULL,dist1="norm",dist2="norm",
#   params.m=params.mF,mcmc=FALSE,mcmc.params=NA,ygrid=ygrid, bsrep=100,
#   n.startvals=30000,dist.para.table=dist.para.t)
#   tsreps1F[r]<-tsF$pval.ks
#   tsreps2F[r]<-tsF$pval.cvm
# }
# time.taken<-Sys.time()-sys.t
# time.taken
# cbind(tsreps1T,tsreps2T,tsreps1F,tsreps2F)
# 
# data(dist.para.t)
# data(params)
# 
# ### example two - Dagum and log-normal - no mcmc
# ##putting list elements from params into matrix form for params.m
# params.m<-matrix(NA,length(params$aft.v),6+4)
# params.m[,1]<-params[[which(names(params)=="bft.v")]]
# params.m[,2]<-params[[which(names(params)=="aft.v")]]
# params.m[,3]<-params[[which(names(params)=="cft.v")]]
# params.m[,4]<-1
# params.m[,5]<-params[[which(names(params)=="mupt.v")]]
# params.m[,6]<-params[[which(names(params)=="sigmapt.v")]]
# params.m[,7]<-params[[which(names(params)=="punemp.v")]]
# params.m[,8]<-params[[which(names(params)=="pft.v")]]
# params.m[,9]<-params[[which(names(params)=="ppt.v")]]
# params.m[,10]<-params[[which(names(params)=="pemp.v")]]
# 
# set.seed(123)
# reps<-30
# tsreps1T<-rep(NA,reps)
# tsreps2T<-rep(NA,reps)
# tsreps1F<-rep(NA,reps)
# tsreps2F<-rep(NA,reps)
# sys.t<-Sys.time()
# for(r in 1:reps){ 
#   ## creates variables under consideration and dimnames
#   n  <- dim(params.m)[1]
#   mcmcsize<-params$mcmcsize
#   ages <- params$ages
#   unems <- params$unems
#   educlvls <- params$educlvls
#   OW <- params$OW
#   ## simulate two samples
#   ages.s <- sample(ages,n,replace=TRUE)
#   unems.s<- sample(unems,n,replace=TRUE)
#   edu.s  <- sample(c(-1,1),n,replac=TRUE)
#   OW.s   <- sample(c(-1,1),n,replac=TRUE)
#   y.sim<-rep(NA,n)
#   p.sel<-sample(1:dim(params.m)[1],n)
#   for(i in 1:n){
#     p<-p.sel[i]
#     #p<-sample(1:n,1) #select a random individual
#     y.sim[i]<-ysample.md(1,"GB2","LOGNO",
#                          theta=c(params$bft.v[p],params$aft.v[p],
#                                  params$cft.v[p],1,
#                                  params$mupt.v[p],params$sigmapt.v[p]),
#                          params$punemp.v[p],params$pft.v[p],params$ppt.v[p],
#                          dist.para.t)
#   }
#   dat<-cbind(y.sim,ages.s,unems.s,edu.s,OW.s)
#   y.simF<- rnorm(n,mean(y.sim),sd(y.sim))
#   y.simF[y.simF<0]<-0
#   datF<-dat
#   datF[,1]<-y.simF
#   ygrid <- seq(0,1e6,by=1000) #quantile(y,taus)
#   ##executing test
#   tsT<-sadr.test(data=dat,y.pos=NULL,dist1="GB2",dist2="LOGNO",params.m=
#                  params.m[p.sel,],mcmc=FALSE,mcmc.params=NA,ygrid=ygrid, 
#                  bsrep=100,n.startvals=30000,dist.para.table=dist.para.t)
#   tsreps1T[r]<-tsT$pval.ks
#   tsreps2T[r]<-tsT$pval.cvm
#   tsF<-sadr.test(data=datF,y.pos=NULL,dist1="GB2",dist2="LOGNO",
#                  params.m=params.m[p.sel,],mcmc=FALSE,mcmc.params=NA,
#                  ygrid=ygrid, 
#                  bsrep=100,n.startvals=30000,dist.para.table=dist.para.t)
#   tsreps1F[r]<-tsF$pval.ks
#   tsreps2F[r]<-tsF$pval.cvm
# }
# time.taken<-Sys.time()-sys.t
# time.taken
# cbind(tsreps1T,tsreps2T,tsreps1F,tsreps2F)
# 
# 
# 
# 
# 
# ### example three - two normals, with mcmc
# set.seed(1234)
# n<-1000 #no of observations
# m<-100 #no of mcmc samples
# sigma<-0.1
# X.theta<-c(1,10,1,10)
# #without weights
# X.gen<-function(n,paras){
#   X<-matrix(c(round(runif(n,paras[1],paras[2])),round(runif(n,paras[3],
#             paras[4]))),ncol=2)
#   return(X)
# }
# X <- X.gen(n,X.theta)
# 
# beta.mu1   <- 1
# beta.sigma1<- 0.1
# beta.mu2   <- 2
# beta.sigma2<- 0.1
# pi0        <- 0.3
# pi01       <- 0.8
# pi1        <- (1-pi0)*pi01
# pi2        <- 1-pi0-pi1
# 
# mcmc.params.a<-array(NA,dim=c(m,n,8))
# mcmc.params.a[,,1]<-(0+beta.mu1+rnorm(m,0,beta.mu1/10))%*%t(X[,1]) 
      #assume sd of mcmc as 10% of parameter value
# mcmc.params.a[,,2]<-(0+beta.sigma1+rnorm(m,0,beta.sigma1/10))%*%t(X[,1]) 
      #must not be negative!, may be for other seed!
# mcmc.params.a[,,3]<-(0+beta.mu2+rnorm(m,0,beta.mu2/10))%*%t(X[,2])
# mcmc.params.a[,,4]<-(0+beta.sigma2+rnorm(m,0,beta.sigma2/10))%*%t(X[,2]) 
      #must not be negative!, may be for other seed!
# mcmc.params.a[,,5]<-(pi0+rnorm(m,0,pi0/10))%*%t(rep(1,n))
# mcmc.params.a[,,8]<-(pi01+rnorm(m,0,pi01/10))%*%t(rep(1,n))
# mcmc.params.a[,,6]<-(1-mcmc.params.a[,,5])*mcmc.params.a[,,8]
# mcmc.params.a[,,7]<-1-mcmc.params.a[,,5]-mcmc.params.a[,,6]
# 
# params.m<-apply(mcmc.params.a,MARGIN=c(2,3),FUN=quantile,probs=0.5)
# 
# mcmc.params.aF<-array(NA,dim=c(m,n,8))
# mcmc.params.aF[,,1]<-(10+beta.mu1+rnorm(m,0,beta.mu1/10))%*%t(X[,1]) 
      #assume sd of mcmc as 10% of parameter value
# mcmc.params.aF[,,2]<-(0+beta.sigma1+rnorm(m,0,beta.sigma1/10))%*%t(X[,1]) 
      #must not be negative!, may be for other seed!
# mcmc.params.aF[,,3]<-(0+beta.mu2+rnorm(m,0,beta.mu2/10))%*%t(X[,2])
# mcmc.params.aF[,,4]<-(2+beta.sigma2+rnorm(m,0,beta.sigma2/10))%*%t(X[,2]) 
      #must not be negative!, may be for other seed!
# mcmc.params.aF[,,5]<-(pi0+rnorm(m,0,pi0/10))%*%t(rep(1,n))
# mcmc.params.aF[,,8]<-(pi01+rnorm(m,0,pi01/10))%*%t(rep(1,n))
# mcmc.params.aF[,,6]<-(1-mcmc.params.aF[,,5])*mcmc.params.aF[,,8]
# mcmc.params.aF[,,7]<-1-mcmc.params.aF[,,5]-mcmc.params.aF[,,6]
# 
# params.mF<-apply(mcmc.params.aF,MARGIN=c(2,3),FUN=quantile,probs=0.5)
# 
# reps<-30
# tsreps1T<-rep(NA,reps)
# tsreps2T<-rep(NA,reps)
# tsreps1F<-rep(NA,reps)
# tsreps2F<-rep(NA,reps)
# sys.t<-Sys.time()
# for(r in 1:reps){
#   Y <- rep(NA,n)
#   for(i in 1:n){
#     Y[i] <- ysample.md(1,dist1,dist2,theta=params.m[i,1:4],params.m[i,5],
#                        params.m[i,6],params.m[i,7],dist.para.t)
#   }  
#   dat<-cbind(Y,X)
#   y.pos<-1
#   ygrid<-seq(min(Y),round(max(Y)*1.2,-1),by=1)  
#   tsT<-sadr.test(data=dat,y.pos=NULL,dist1="norm",dist2="norm",params.m=
#                  params.m,mcmc=TRUE,mcmc.params=mcmc.params.a,ygrid=ygrid, 
#                  bsrep=100,n.startvals=30000,dist.para.table=dist.para.t)
#   tsreps1T[r]<-tsT$pval.ks
#   tsreps2T[r]<-tsT$pval.cvm
#   tsF<-sadr.test(data=dat,y.pos=NULL,dist1="norm",dist2="norm",
#                  params.m=params.mF,mcmc=TRUE,mcmc.params=mcmc.params.aF,
#                  ygrid=ygrid, bsrep=100,n.startvals=30000,
#                  dist.para.table=dist.para.t)
#   tsreps1F[r]<-tsF$pval.ks
#   tsreps2F[r]<-tsF$pval.cvm
#   #c(ts$teststat.ks,ts$teststat.cvm)
#   #c(ts$pval.ks,ts$pval.cvm)
#   
# }
# time.taken<-Sys.time()-sys.t
# time.taken
# cbind(tsreps1T,tsreps2T,tsreps1F,tsreps2F)
# 
# 
# 
# ### example four - two normals, with mcmc and slight deviation from truth 
#     in true params
# library(acid)
# data(dist.para.t)
# data(params)
# dist1<-"norm"
# dist2<-"norm"
# 
# set.seed(1234)
# n<-1000 #no of observations
# m<-100 #no of mcmc samples
# sigma<-0.1
# X.theta<-c(1,10,1,10)
# #without weights
# X.gen<-function(n,paras){
#   X<-matrix(c(round(runif(n,paras[1],paras[2])),round(runif(n,paras[3],
#             paras[4]))),ncol=2)
#   return(X)
# }
# X <- X.gen(n,X.theta)
# 
# beta.mu1   <- 1
# beta.sigma1<- 0.1
# beta.mu2   <- 2
# beta.sigma2<- 0.1
# pi0        <- 0.3
# pi01       <- 0.8
# pi1        <- (1-pi0)*pi01
# pi2        <- 1-pi0-pi1
# 
# mcmc.params.a<-array(NA,dim=c(m,n,8))
# mcmc.params.a[,,1]<-(beta.mu1/10+beta.mu1+rnorm(m,0,beta.mu1/10))%*%t(X[,1]) 
       #assume sd of mcmc as 10% of parameter value
# mcmc.params.a[,,2]<-(0+beta.sigma1+rnorm(m,0,beta.sigma1/10))%*%t(X[,1]) 
       #must not be negative!, may be for other seed!
# mcmc.params.a[,,3]<-(0+beta.mu2+rnorm(m,0,beta.mu2/10))%*%t(X[,2])
# mcmc.params.a[,,4]<-(beta.sigma2/10+beta.sigma2+rnorm(m,0,
#                      beta.sigma2/10))%*%t(X[,2]) 
       #must not be negative!, may be for other seed!
# mcmc.params.a[,,5]<-(pi0+rnorm(m,0,pi0/10))%*%t(rep(1,n))
# mcmc.params.a[,,8]<-(pi01+rnorm(m,0,pi01/10))%*%t(rep(1,n))
# mcmc.params.a[,,6]<-(1-mcmc.params.a[,,5])*mcmc.params.a[,,8]
# mcmc.params.a[,,7]<-1-mcmc.params.a[,,5]-mcmc.params.a[,,6]
# 
# params.m<-apply(mcmc.params.a,MARGIN=c(2,3),FUN=quantile,probs=0.5)
# 
# mcmc.params.aF<-array(NA,dim=c(m,n,8))
# mcmc.params.aF[,,1]<-(10+beta.mu1+rnorm(m,0,beta.mu1/10))%*%t(X[,1]) 
       #assume sd of mcmc as 10% of parameter value
# mcmc.params.aF[,,2]<-(0+beta.sigma1+rnorm(m,0,beta.sigma1/10))%*%t(X[,1]) 
       #must not be negative!, may be for other seed!
# mcmc.params.aF[,,3]<-(0+beta.mu2+rnorm(m,0,beta.mu2/10))%*%t(X[,2])
# mcmc.params.aF[,,4]<-(2+beta.sigma2+rnorm(m,0,beta.sigma2/10))%*%t(X[,2]) 
       #must not be negative!, may be for other seed!
# mcmc.params.aF[,,5]<-(pi0+rnorm(m,0,pi0/10))%*%t(rep(1,n))
# mcmc.params.aF[,,8]<-(pi01+rnorm(m,0,pi01/10))%*%t(rep(1,n))
# mcmc.params.aF[,,6]<-(1-mcmc.params.aF[,,5])*mcmc.params.aF[,,8]
# mcmc.params.aF[,,7]<-1-mcmc.params.aF[,,5]-mcmc.params.aF[,,6]
# 
# params.mF<-apply(mcmc.params.aF,MARGIN=c(2,3),FUN=quantile,probs=0.5)
# 
# reps<-30
# tsreps1T<-rep(NA,reps)
# tsreps2T<-rep(NA,reps)
# tsreps1F<-rep(NA,reps)
# tsreps2F<-rep(NA,reps)
# sys.t<-Sys.time()
# for(r in 1:reps){
#   Y <- rep(NA,n)
#   for(i in 1:n){
#     Y[i] <- ysample.md(1,dist1,dist2,theta=params.m[i,1:4],params.m[i,5],
#                        params.m[i,6],params.m[i,7],dist.para.t)
#   }
#   
#   dat<-cbind(Y,X)
#   y.pos<-1
#   ygrid<-seq(min(Y),round(max(Y)*1.2,-1),by=1)  
#   tsT<-sadr.test(data=dat,y.pos=NULL,dist1="norm",dist2="norm",
#                  params.m=params.m,mcmc=TRUE,mcmc.params=mcmc.params.a,
#                  ygrid=ygrid, bsrep=100,n.startvals=30000,
#                  dist.para.table=dist.para.t)
#   tsreps1T[r]<-tsT$pval.ks
#   tsreps2T[r]<-tsT$pval.cvm
#   tsF<-sadr.test(data=dat,y.pos=NULL,dist1="norm",dist2="norm",
#                  params.m=params.mF,mcmc=TRUE,mcmc.params=mcmc.params.aF,
#                  ygrid=ygrid, bsrep=100,n.startvals=30000,
#                  dist.para.table=dist.para.t)
#   tsreps1F[r]<-tsF$pval.ks
#   tsreps2F[r]<-tsF$pval.cvm
#   #c(ts$teststat.ks,ts$teststat.cvm)
#   #c(ts$pval.ks,ts$pval.cvm)
#   
# }
# time.taken<-Sys.time()-sys.t
# time.taken
# cbind(tsreps1T,tsreps2T,tsreps1F,tsreps2F)

Standard Deviation of the Generalised Beta Distribution of Second Kind

Description

This function calculates the standard deviation of the Generalised Beta Distribution of Second Kind.

Usage

sd.GB2(b, a, p, q)

Arguments

b

the parameter b of the Dagum distribution as defined by Kleiber and Kotz (2003).

a

the parameter a of the Dagum distribution as defined by Kleiber and Kotz (2003).

p

the parameter p of the Dagum distribution as defined by Kleiber and Kotz (2003).

q

the parameter q of the Dagum distribution as defined by Kleiber and Kotz (2003).

Value

returns the standard deviation.

Author(s)

Alexander Sohn

References

Kleiber, C. and Kotz, S. (2003): Statistical Size Distributions in Economics and Actuarial Sciences, Wiley, Hoboken.

Examples

a.test<- 4
b.test<- 20000
p.test<- 0.7
q.test<- 1
alpha.test<-1
GB2sample<-rGB2(10000,b.test,a.test,p.test,q.test)
sd.GB2(b.test,a.test,p.test,q.test)
sd(GB2sample)

Single-parameter Gini Coefficient

Description

This function computes the Single-parameter Gini coefficient (a.k.a. generalised Gini coefficient or extended Gini coefficient) for a vector of observations.

Usage

sgini(x, nu = 2, w = NULL)

Arguments

x

a vector of observations.

nu

a scalar entailing the parameter that tunes the degree of the policy maker's aversion to inequality. See Yaari, 1988 for details.

w

a vector of weights.

Value

Gini

the Gini coefficient for the sample.

bcGini

the bias-corrected Gini coefficient for the sample.

Author(s)

Alexander Sohn

References

van Kerm, P. (2009): sgini - Generalized Gini and Concentration coefficients (with factor decomposition) in Stata', CEPS/INSTEAD, Differdange, Luxembourg.

Yaari, M.E. (1988): A Controversal Proposal Concerning Inequality Measurement, Journal of Economic Theory, Vol. 44, pp. 381-397.

Examples

set.seed(123)
x <- rnorm(100,10,1)
gini(x)$Gini
sgini(x,nu=2)$Gini

Single-parameter Gini Coefficient for an Income Distribution

Description

This function approximates the Single-parameter Gini coefficient for a distribution specified by a vector of densities and a corresponding income vector. A point mass at zero is allowed.

Usage

sgini.den(incs, dens, nu = 2, pm0 = NA, lower = NULL, upper = NULL)

Arguments

incs

a vector with sorted income values.

dens

a vector with the corresponding densities.

nu

a scalar entailing the parameter that tunes the degree of the policy maker's aversion to inequality. See Yaari, 1988 for details.

pm0

the point mass for zero incomes. If not specified no point mass is assumed.

lower

the lower bound of the income range considered.

upper

the upper bound of the income range considered.

Value

Gini

the approximation of the Gini coefficient.

pm0

the point mass for zero incomes used.

lower

the lower bound of the income range considered used.

upper

the upper bound of the income range considered used.

Author(s)

Alexander Sohn

References

van Kerm, P. (2009): sgini - Generalized Gini and Concentration coefficients (with factor decomposition) in Stata', CEPS/INSTEAD, Differdange, Luxembourg.

Yaari, M.E. (1988): A Controversal Proposal Concerning Inequality Measurement, Journal of Economic Theory, Vol. 44, pp. 381-397.

Examples

## without point mass
set.seed(123)
x <- rnorm(1000,10,1)
incs <- seq(1,20,length.out=1000)
dens <- dnorm(incs,10,1)
lower=NULL;upper=NULL;pm0<-NA
gini(x)$Gini
sgini(x,nu=2)$Gini
sgini.den(incs,dens)$Gini


## with point mass
set.seed(123)
x <- c(rep(0,1000),rnorm(1000,10,1))
incs <- c(0,seq(1,20,length.out=1000))
dens <- c(0.5,dnorm(incs[-1],10,1)/2)
gini(x)$Gini
sgini(x,nu=2)$Gini
sgini.den(incs,dens,pm = 0.5)$Gini

Skewness of the Generalised Beta Distribution of Second Kind

Description

This function calculates the skewness of the Generalised Beta Distribution of Second Kind.

Usage

skewness.GB2(b, a, p, q)

Arguments

b

the parameter b of the Dagum distribution as defined by Kleiber and Kotz (2003).

a

the parameter a of the Dagum distribution as defined by Kleiber and Kotz (2003).

p

the parameter p of the Dagum distribution as defined by Kleiber and Kotz (2003).

q

the parameter q of the Dagum distribution as defined by Kleiber and Kotz (2003).

Value

returns the skewness.

Author(s)

Alexander Sohn

References

Kleiber, C. and Kotz, S. (2003): Statistical Size Distributions in Economics and Actuarial Sciences, Wiley, Hoboken.

Examples

a.test<- 4
b.test<- 20000
p.test<- 0.7
q.test<- 1
alpha.test<-1
GB2sample<-rGB2(10000,b.test,a.test,p.test,q.test)
skewness.GB2(b.test,a.test,p.test,q.test)
#require(e1071)
#skewness(GB2sample)#note that this estimation is highly unstable even for larger sample sizes.

Theil Index for the Gamma Distribution

Description

This function computes the Theil index for the gamma distribution.

Usage

theil.gamma(p)

Arguments

p

the shape parameter p of the gamma distribution as defined by Kleiber and Kotz (2003).

Value

returns the Theil index.

Author(s)

Alexander Sohn

References

Cowell, F.A. (2000): Measurement of Inequality, in: Atkinson and Bourguignon (eds.), Handbook of Income Distribution, pp. 1-86, Elsevier, Amsterdam.

Kleiber, C. and Kotz, S. (2003): Statistical Size Distributions in Economics and Actuarial Sciences, Wiley, Hoboken.

Examples

shape.test <- 5
scale.test  <- 50000
y <- rgamma(10000,shape=shape.test,scale=scale.test)
entropy(y,1)
theil.gamma(shape.test)

Atkinson Inequality Index

Description

This function computes the Atkinson inequality index for a vector of observations with corresponding weights.

Usage

weighted.atkinson(x, w = NULL, epsilon = 1, wscale = 1000)

Arguments

x

a vector of observations.

w

a vector of weights. If

epsilon

inequality aversion parameter as denoted by Atkinson (1970). The default is epsilon=1.

wscale

a scale by which the weights are adjusted such that can be rounded to natural numbers.

Value

returns the selected Atkinson inequality index.

Author(s)

Alexander Sohn

References

Atkinson, A.B. (1970): On the Measurment of Inequality, in: Journal of Economic Theory, Vol. 2(3), pp. 244-263.

Examples

x <- c(541, 1463, 2445, 3438, 4437, 5401, 6392, 8304, 11904, 22261)
w <- sample(1:2,length(x),replace=TRUE)
weighted.atkinson(x,w)

Coefficient of Variation

Description

This function computes the Coefficient of Variation for a vector of observations and corresponding weights.

Usage

weighted.coeffvar(x, w)

Arguments

x

a vector of observations.

w

a vector of weights.

Value

cv

returns the coefficient of variation without bias correction.

bccv

returns the coefficient of variation with bias correction.

Warning

Weighting is not properly accounted for in the sample adjustment of bccv!

Author(s)

Alexander Sohn

References

Atkinson, A.B. and Bourguignon, F. (2000): Income Distribution and Economics, in: Atkinson and Bourguignon (eds.), Handbook of Income Distribution, pp. 1-86, Elsevier, Amsterdam.

Examples

# generate vector (of incomes)
x <- c(541, 1463, 2445, 3438, 4437, 5401, 6392, 8304, 11904, 22261)
w <- sample(1:10,length(x), replace=TRUE)
weighted.coeffvar(x,w)

Measures of the Generalised Entropy Family

Description

This function computes the Measures of the Generalised Entropy Family for a vector of observations with corresponding weights.

Usage

weighted.entropy(x, w = NULL, alpha = 1)

Arguments

x

a vector of observations.

w

a vector of weights.

alpha

the parameter for the generalised entropy family of measures, denoted by alpha by Cowell (2000). Note that this parameter notation differs from the notation used in the ineq package.

Value

returns the entropy measure.

Author(s)

Alexander Sohn

References

Cowell, F.A. (2000): Measurement of Inequality, in: Atkinson and Bourguignon (eds.), Handbook of Income Distribution, pp. 1-86, Elsevier, Amsterdam.

Examples

# generate vector (of incomes)
x <- c(541, 1463, 2445, 3438, 4437, 5401, 6392, 8304, 11904, 22261)
w <- sample(1:2,length(x),replace=TRUE)
weighted.entropy(x,w)

Gini Coefficient

Description

This function computes the Gini coefficient for a vector of observations with corresponding weights.

Usage

weighted.gini(x, w = NULL)

Arguments

x

a vector of observations.

w

a vector of weights.

Value

returns the Gini coefficient.

Author(s)

Alexander Sohn

References

Cowell, F.A. (2000): Measurement of Inequality, in: Atkinson and Bourguignon (eds.), Handbook of Income Distribution, pp. 1-86, Elsevier, Amsterdam.

Examples

# generate vector (of incomes)
x <- c(541, 1463, 2445, 3438, 4437, 5401, 6392, 8304, 11904, 22261)
w <- sample(1:2,length(x),replace=TRUE)
weighted.gini(x,w)

Moments of a Random Variable

Description

This functions calculates the first three moments as well as mean, standard deviation and skewness for a vector of observations with corresponding weights.

Usage

weighted.moments(x, w8 = NULL)

Arguments

x

a vector of observations.

w8

a vector of weights.

Value

fm

returns the first moment.

weighted.mean

returns the mean.

sm

returns the second moment.

weighted.sd

returns the uncorrected (population) standard deviation.

wtd.sd

returns the sample-size corrected standard deviation estimate.

tm

returns the third moment.

w.skew.SAS

returns the skewness estimate as implemented in SAS.

w.skew.Stata

returns the skewness estimate as implemented in Stata.

Author(s)

Alexander Sohn

Sampling Incomes from a Mixture of Income Distributions

Description

This function samples incomes from a mixture of two continuous income distributions and a point mass for zero-incomes.

Usage

ysample.md(n, dist1, dist2, theta, p0, p1, p2, dist.para.table)

Arguments

n

number of observations.

dist1

dist2

theta

vector with the parameters of dist1 and dist2. Order must be the same as in the functions for the distributions.

p0

scalar with probability mass for the point mass.

p1

scalar with probability mass for dist1.

p2

scalar with probability mass for dist2.

dist.para.table

a table of the same form as dist.para.t with distribution name, function name and number of parameters.

Value

returns the sample of observations.

Author(s)

Alexander Sohn

Examples

data(dist.para.t)
ygrid<-seq(0,1e5,by=1000)
theta<-c(5,1,10,3)
p0<-0.2
p1<-0.3
p2<-0.5
n <-10
ysample.md(n, "LOGNO", "LOGNO", theta, p0, p1, p2, dist.para.t)

Analysing Conditional Income Distributions

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Mean of the Generalised Beta Distribution of Second Kind

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Atkinson Inequality Index

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Atkinson Index for a Generalised Beta Distribution of Second Kind

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Atkinson Index for an Income Distribution

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Atkinson Index for a Mixture of Income Distributions

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Cumulative Density Function of Log-Normal Mixture Distribution

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Cumulative Density Function of Dagum Mixture Distribution

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Coefficient of Variation

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Simultaneous Confidence Bands

Description