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 $\Omega$.
"Rational Variance Functions." Ann. Statist. 17 (2) 741 - 748, June, 1989. https://doi.org/10.1214/aos/1176347139