ON MODALITY AND DIVISIBILITY OF POISSON AND BINOMIAL MIXTURES

Abstract

Some structural aspects of mixtures, in general, have been previously investigated by the author in [I] and [2]. The aim of this article is to investigate some important structural properties of the special cases of Poisson and binomial mixtures in detail. Some necessary and sufficient conditions are arrived at for different modality and divisibility properties of a Poisson mixture based on similar properties of its mixing distribution. These simplify and improve the results of Holgate [7], Shanbhagh & Westcott [12] and Forst [6] in this connection. Whenever the preservation of a property of the mixing distribution is not held by the mixture, proper counterexamples are provided to illustrate the point. Then, a similar argument is employed to extend these ideas analogously to the case of binomial mixtures.