R: Kemp and Kemp generalized hypergeometric types

ghyper.types {SuppDists} | R Documentation

## Kemp and Kemp generalized hypergeometric types

### Description

Generalized hypergeometric types as given by Kemp and Kemp

### Two-way table

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.

### Kemp and Kemp types

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 …

### Author(s)

Bob Wheeler bwheelerg@gmail.com

### References

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

[Package SuppDists version 1.1-9.4 Index]