Become an expert in R — Interactive courses, Cheat Sheets, certificates and more!
Get Started for Free

ghyperTypes

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

References

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


SuppDists

Supplementary Distributions

v1.1-9.5
GPL (>= 2)
Authors
Bob Wheeler [aut], Thorsten Pohlert [ctb, cre] (<https://orcid.org/0000-0003-3855-3025>)
Initial release
2020-01-13

We don't support your browser anymore

Please choose more modern alternatives, such as Google Chrome or Mozilla Firefox.