On quasi Pólya thinning operator

Abstract

Thinning operation is a stochastic operation that shrinks a random count variable into a smaller one. This kind of random operation has been intensively studied during the seventies to characterize some count distributions, such as the Poisson distribution using the binomial thinning operator (also named binomial damage model). Then, the closure under thinning operator has been studied in order to define some classes of integer valued autoregressive (INAR) models for count time series. These two properties will be studied in this paper for the new class of quasi Pólya thinning operators. Classical results concerning the binomial thinning operator are recovered as a special case. The quasi Pólya thinning operator is related to the new class of quasi Pólya splitting distributions, defined for multivariate count data. The probabilistic graphical model (PGM) of these multivariate distributions is characterized. Finally a general class of integer valued autoregressive models is introduced, including the usual cases of Poisson marginal or generalized Poisson as a special cases and the generalized negative binomial as a new case.