Rational Variance Functions

Abstract

Exponential dispersion models play an important role in the context of generalized linear models, where error distributions, other than the normal, are considered. Any statistical model expressible in terms of a variance-mean relation $(V, \Omega)$ leads to an exponential dispersion model provided that $(V, \Omega)$ is a variance function of a natural exponential family: Here $\Omega$ is the domain of means and $V$ is the variance function of the natural exponential family. Therefore, it is of a particular interest to examine whether a pair $(V, \Omega)$ can serve as the variance function of a natural exponential family. In this study we consider the case where $\Omega$ is bounded and examine whether $V$ can be the restriction to $\Omega$ of a rational function vanishing at the boundary points of $\Omega$. The class of such functions is large and contains the important subclass of polynomials. It is shown that, apart from the binomial family (possessing a quadratic variance function) and affine transformations thereof, there exists no natural exponential family with variance function belonging to this class. Such a result implies, in particular, that the only variance functions of natural exponential families among polynomials of at least third degree are those restricted to unbounded domains $\Omega$.