ghyper.types: Kemp and Kemp generalized hypergeometric types

Description

Generalized hypergeometric types as given by Kemp and Kemp

The basic representation is in terms of a two-way table:

and the associated hypergeometric probability \(P(x)=C_x^a C_{k-x}^b / C_k^N\).

The types are classified according to ranges of a, k, and N.

Minor modifications in the definition of three of the types have been made to avoid numerical difficulties. Note, J denotes a nonnegative integer.

[Classic]
	\(0<a, 0<N, 0<k\)
	integers: a, N, k.
	\(max(0,a+k-N) \le x \le min(a,k)\)
[IA(i)] (Real classic)	at least one noninteger parameter
	\(0<a, 0<N, 0<k, k-1<a<N-(k-1)\)
	integer: k
	\( 0 \le x \le a\)
[IA(ii)] (Real classic)	at least one noninteger parameter
	\(0<a, 0<N, 0<k, a-1<k<N-(a-1)\)
	integer: a
	\(0 \le x \le a\)
	Interchanging a and k transforms this to type IA(i)
[IB]
	\(0<a, 0<N, 0<k, a+k-1<N, J < (a,k) < J+1\)
	integer: \(0 \le J\)
	non-integer: a, k
	\(0 <= x \dots \)
	NOTE: Kemp and Kemp specify \(-1<N\).
	No practical applications for this distribution.
[IIA] (negative hypergeometric)
	\(a<0, N<a-1,0<k\)
	integer: k
	\(0 \le x \le k\)
	NOTE: Kemp and Kemp specify \(N<a, N \ne a-1\)
[IIB]
	\(a<0, -1<N<k+a-1, 0<k, J < (k,k+a-1-N) < J+1\)
	non-integer: k
	integer: \(0 \le J\)
	\(0 \le x ....\)
	This is a very strange distribution. Special calculations were used.
	Note: No practical applications.
[IIIA] (negative hypergeometric)
	\(0<a,N<k-1,k<0\)
	integer: a
	\(0 \le x \le a\)
	Interchanging a and k transforms this to type IIA
	NOTE: Kemp and Kemp specify \(N<k, N \ne k-1\)
[IIIB]
	\(0<a,-1<N<a+k-1,k<0, J<(a,a+k-1-N)<J+1\)
	non integer: a
	integer: \(0 \le J\)
	\(0 \le x \dots \)
	Interchanging a and k transforms this to type IIB
	Note: No practical applications
[IV] (Generalized Waring)
	\(a<0,-1<N, k<0\)
	\(0 \le x \dots\)

Bob Wheeler

Kemp, C.D., and Kemp, A.W. (1956). Generalized hypergeometric distributions. Jour. Roy. Statist. Soc. B. 18. 202-211. 39. 887-895.