Kemp and Kemp generalized hypergeometric types
Generalized hypergeometric types as given by Kemp and Kemp
The basic representation is in terms of a two-way table:
| x | k-x | k |
| a-x | b-k+x | N-k |
| a | b | N |
and the associated hypergeometric probability P(x)=choose(a, x) choose(b, k-x) / choose(N, k).
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) <= x <= 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 <= x <= k | |
| [IA(ii)] (Real classic) | at least one noninteger parameter |
| 0<a, 0<N, 0<k, a-1<k<N-(a-1) | |
| integer: a | |
| 0 <= x <= 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 <= J | |
| non-integer: a, k | |
| 0 <= x … | |
| 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 <= x <= k | |
| NOTE: Kemp and Kemp specify N<a and N!=(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 <= J | |
| 0 <= 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 <= x <= a | |
| Interchanging a and k transforms this to type IIA | |
| NOTE: Kemp and Kemp specify N<k and N != 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 <= J | |
| 0 <= x … | |
| Interchanging a and k transforms this to type IIB | |
| Note: No practical applications | |
| [IV] (Generalized Waring) | |
| a<0,-1<N, k<0 | |
| 0 <= x … |
Bob Wheeler
Kemp, C.D., and Kemp, A.W. (1956). Generalized hypergeometric distributions. Jour. Roy. Statist. Soc. B. 18. 202-211. 39. 887-895.
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