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.