POWER SERIES DISTRIBUTIONS IN MATHEMATICAL STATISTICS AND APPLIED PROBABILITY

Discrete distributions, such as the Poisson, having the property that their probability mass functions are proportional to the terms of certain power series functions, are called power series distributions (PSD's).;Under mild conditions, it is shown that every PSD corresponds to a counting process as well as to a birth process. The Poisson process is characterized as being the only renewal process in the class of PSD's. Weighted geometric models are examined and characterizations through the size-biased model of some PSD's are obtained using form-invariance with given displacement. The independent of X and Y in the additive damage model (Z,X,Y) is shown to be a characteristic property of the PSD's. Several basic properties of the series defining functions as well as mathematical operations on the series defining functions and their stochastic interpretations are examined.;The concepts of conjugate pairs (X,O), where X is a discrete rv and O is a continuous rv, are introduced. Their classifications and interrelations are discussed. Characterizations of some PSD's are obtained as unique conjugate pairs. A condition for a mixture on the series parameter of a PSD (having support on the non-negative integers) to be identifiable is given, and is used to characterize the gamma and the beta distributions. Finally, characterizations of the prior distributions through certain forms of the posterior estimate are obtained, and examples are given.;We study distribution theory relating PSD's to absolutely continuous distributions using interesting probabilistic arguments. The upper (lower) tail probabilities of the PSD's are given as lower (upper) tail probabilities of a family of absolutely continuous distributions. These tail probability relationships are extended to cases involving differences of independent rv's having PSD's and non-central absolutely continuous distributions. Monotonicity of the power functions of certain tests based on these non-central distributions with respect to nuisance and non-centrality parameters is considered.