Compound {GB2}

Compound Distribution based on the Generalized Beta Distribution of the Second Kind

Description

Mixture distribution based on the compounding property of the GB2, in short "compound GB2". Decomposition of the GB2 distribution with respect to the left and right tail of the distribution. Calculation of the component densities and cumulative distribution functions. Calculation of the compound density function and the compound cumulative distribution function.

Usage

fg.cgb2(x, shape1, scale, shape2, shape3, pl0, decomp="r")
dl.cgb2(x, shape1, scale, shape2, shape3, pl0, decomp="r") 
pl.cgb2(y, shape1, scale, shape2, shape3, pl0, decomp="r", tol=1e-05)
dcgb2(x, shape1, scale, shape2, shape3, pl0, pl, decomp="r")
pcgb2(y, shape1, scale, shape2, shape3, pl0, pl, decomp="r")
prcgb2(y1, y2, shape1, scale, shape2, shape3, pl0, pl, decomp="r", tol=1e-08, 
debug=FALSE)

Arguments

x

numeric; can be a vector. The value(s) at which the compound density and the component densities are calculated, x is positive.

y

numeric; can be a vector. The value(s) at which the compound distribution function and the component distribution functions are calculated.

y1, y2

numeric values.

shape1, scale, shape2, shape3

numeric; positive parameters of the GB2 distribution.

pl0

numeric; a vector of initial proportions defining the number of components and the weight of each component density in the decomposition. Sums to one.

pl

numeric; a vector of fitted proportions. Sums to one. If pl is equal to pl0, we obtain the GB2 distribution.

decomp

string; specifying if the decomposition of the GB2 is done with respect to the right tail ("r") or the left tail ("l") of the distribution. By default, decomp = "r" - right tail decomposition.

debug

logical; By default, debug = FALSE.

tol

numeric; tolerance with default 0, meaning to iterate until additional terms do not change the partial sum.

Details

The number of components L is given by the length of the vector pl0. In our examples L=3. Let N denote the length of the vector x. Function fg.cgb2 calculates the L gamma factors which multiply the GB2 density in order to obtain the component density f_\ell. These component densities are calculated using the function dl.cgb2. Function pl.cgb2 calculates the corresponding L cumulative component distribution functions. Function dcgb2 calculates the resulting compound density function. Function pcgb2 calculates the compound cumulative distribution function for a vector of values y and function prcgb2, given 2 arguments y1 and y2, calculates the probability P(min(y1,y2) < Y < max(y1,y2)), where the random variable Y follows a compound GB2 distribution.

Value

fg.cgb2 returns a matrix of size N \times L of the Gamma factors, dl.cgb2 returns a matrix of size N \times L of component densities, pl.cgb2 returns a matrix containing the L component cdfs, dcgb2 returns a matrix of size N \times 1 of the GB2 compound density function, pcgb2 returns a matrix of size N \times 1 of the GB2 compound distribution function and prcgb2 returns a probability between 0 and 1.

Author(s)

Monique Graf and Desislava Nedyalkova

References

Graf, M., Nedyalkova, D., Muennich, R., Seger, J. and Zins, S. (2011) AMELI Deliverable 2.1: Parametric Estimation of Income Distributions and Indicators of Poverty and Social Exclusion. Technical report, AMELI-Project.

Examples

#\dontrun{

#\library{cubature}

# GB2 parameters
af <- 5
bf <- 20000
pf <- 0.45 
qf <- 0.75

p0 <- rep(1/3,3)
p1 <- c(0.37,0.43,0.2)

# a vector of values
x <- rep(20000*seq(1,2,length.out=9))

#Gamma components
fg.cgb2(20000,af,bf,pf,qf,p0)
fg.cgb2(Inf,af,bf,pf,qf,p0,"l")

#Component densities
dl.cgb2(x,af,bf,pf,qf,p0)
dl.cgb2(20000,af,bf,pf,qf,p0,"l")

#Component cdf
pl.cgb2(25000,af,bf,pf,qf,p0)

#Compound cdf
pcgb2(x,af,bf,pf,qf,p0,p1)
prcgb2(37000,38000,af,bf,pf,qf,p0,p1,"l")
#}