Thomae {GB2}

Maximum Excess Representation of a Generalized Hypergeometric Function Using Thomae's Theorem

Description

Defines Thomae's arguments from the upper (U) and lower (L) parameters of a _{3}F_{2}(U,L;1). Computes the optimal combination leading to the maximum excess. Computes the optimal combination of Thomae's arguments and calculates the optimal representation of the _{3}F_{2}(U,L;1) using the genhypergeo_series function from package hypergeo. Computes the Gini coefficient for the GB2 distribution, using Thomae's theorem.

Usage

ULg(U, L)
combiopt(g)
Thomae(U, L, lB, tol, maxiter, debug)
gb2.gini(shape1, shape2, shape3, tol=1e-08, maxiter=10000, debug=FALSE)

Arguments

U

numeric; vector of length 3 giving the upper arguments of the generalized hypergeometric function _{3}F_{2}.

L

numeric; vector of length 2 giving the lower arguments of the generalized hypergeometric function _{3}F_{2}.

g

numeric; vector of Thomae's permuting arguments.

lB

numeric; ratio of beta functions (a common factor in the expression of the Gini coefficient under the GB2).

shape1

numeric; positive parameter.

shape2, shape3

numeric; positive parameters of the Beta distribution.

tol

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

maxiter

numeric; maximum number of iterations to perform.

debug

boolean; if TRUE, returns the list of changes to the partial sum.

Details

Internal use only. More details can be found in Graf (2009).

Value

ULg returns a list containing Thomae's arguments and the excess, combiopt gives the optimal combination of Thomae's arguments, Thomae returns the optimal representation of the _{3}F_{2}(U,L;1), gb2.gini returns the value of the Gini coefficient under the GB2.

Author(s)

Monique Graf

References

Graf, M. (2009) An Efficient Algorithm for the Computation of the Gini coefficient of the Generalized Beta Distribution of the Second Kind. ASA Proceedings of the Joint Statistical Meetings, 4835–4843. American Statistical Association (Alexandria, VA).

McDonald, J. B. (1984) Some generalized functions for the size distribution of income. Econometrica, 52, 647–663.

See Also

genhypergeo_series, gini.gb2