Combination of Reproductive Models

Abstract

Suppose $s$ is a random variate that follows a statistical model with parameter $\omega$, and let $s_1, s_2, \cdots, s_n, \cdots$ be independent and identically distributed observations of $s$. The model is reproductive in $s$ and $\omega$ if for any $n$ the mean $\bar{s} = (s_1 + \cdots + s_n)/n$ follows the same model as $s$ but with parameter $n\omega$ instead of $\omega$. Suitable combinations of reproductive models yield reproductive models for higher-dimensional variates. This combination technique is discussed and illustrated by examples. It is possible, in particular, to construct reproductive combinations of gamma, inverse-Gaussian and Gaussian distributions, determined by a regression structure, which may conveniently be described in graph-theoretic terms. The graph-theoretical interpretation makes it feasible to draw conclusions about conditional independencies in the models concerned, by means of a very general result for Markovian-type probability laws on graphs due to Kiiveri, Speed and Carlin (1984). Most of the models discussed are exponential, of a form, which in conjunction with the reproductivity, implies various useful distributional properties, derivable from the general theory of reproductive exponential models.