Two Generalizations of the Binomial Distribution

Patricia M. E. Altham

Journal of the Royal Statistical Society Series C, 1978, vol. 27, issue 2, 162-167

Abstract: The sum of k independent and identically distributed (0, 1) variables has a binomial distribution. If the variables are identically distributed but not independent, this may be generalized to a two‐parameter distribution where the k variables are assumed to have a symmetric joint distribution with no second‐ or higher‐order “interactions”. Two distinct generalizations are obtained, depending on whether the “multiplicative” or “additive” definition of “interaction” for discrete variables is used. The multiplicative generalization gives rise to a two‐parameter exponential family, which naturally includes the binomial as a special case. Whereas with a beta‐binomially distributed variable the variance always exceeds the corresponding binomial variance, the “additive” or “multiplicative” generalizations allow the variance to be greater or less than the corresponding binomial quantity. The properties of these two distributions are discussed, and both distributions are fitted, successfully, to data given by Skellam (1948) on the secondary association of chromosomes in Brassica.

Date: 1978
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