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June, 1989 Rational Variance Functions
Shaul K. Bar-Lev, Daoud Bshouty
Ann. Statist. 17(2): 741-748 (June, 1989). DOI: 10.1214/aos/1176347139


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 $\Omega$.


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Shaul K. Bar-Lev. Daoud Bshouty. "Rational Variance Functions." Ann. Statist. 17 (2) 741 - 748, June, 1989.


Published: June, 1989
First available in Project Euclid: 12 April 2007

zbMATH: 0672.62023
MathSciNet: MR994264
Digital Object Identifier: 10.1214/aos/1176347139

Primary: 62E10
Secondary: 62J02

Keywords: entire function , exponential dispersion model , meromorphic function , Natural exponential family , rational variance function , variance function

Rights: Copyright © 1989 Institute of Mathematical Statistics


Vol.17 • No. 2 • June, 1989
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