Generalized Hypergeometric Distributions

Summary An examination is made of the conditions under which the hypergeometric series provides discrete frequency distributions; eight sets of conditions exist. Urn model representations, other derivations and special forms are discussed. The limiting forms are positive and negative binomials and the Poisson distribution. Other special forms, besides the familiar case of sampling without replacement from a finite population, are the distributions described by Skellam and Irwin. Illustrative examples are given. The existence and nature of the factorial moments is considered for each of the eight sets of conditions—under some circumstances distributions exist for which there are no finite moments. Formulae are given for the finite moments and for the indices of dispersion and skewness; these indices can take a very wide range of values. Moment and maximum likelihood estimation of the parameters is also mentioned; for some forms this involves the problem of estimating restricted parameters.