VARIANCE FUNCTIONS WITH MEROMORPHIC MEANS

• 成果类型: Article
• 署名作者: BARLEV, SK; BSHOUTY, D; ENIS, P
• 署名单位: Technion Israel Institute of Technology; State University of New York (SUNY) System; University at Buffalo, SUNY
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/aop/1176990348
• 发表日期: 1991
• 页码: 1349-1366
• 关键词: natural exponential-families; dispersion models

摘要

A natural exponential family F is characterized by the pair (V, OMEGA), called the variance function (VF), where OMEGA is the mean domain and V is the variance of F expressed in terms of the mean. Any VF can be used to construct an exponential dispersion model, thus providing a potential generalized linear model. A problem of increasing interest in the literature is the following: Given an open interval OMEGA and a function V defined on OMEGA, is the pair (V, OMEGA) a VF of a natural exponential family? In this paper, we develop a complex analytic approach to this question and focus on VF's having meromorphic mean functions; that is, if T is the Laplace transform of an element of the family, then T'/T is extendable to a meromorphic function on C. We derive properties of such VF's and characterize a class of VF's (V, OMEGA), where V admits a unique analytic continuation in C, except for isolated singularities. (Included in this class are VF's having V's that admit meromorphic continuation to C.) We show that this class equals the set of VF's which are at most second degree polynomials. We also investigate the class in which V has the form P + Q square-root R, where P and Q are arbitrary rational functions and R is a polynomial of at most second degree. We characterize all VF's in this class for which the mean function is meromorphic and show that P = kR for some constant k and Q is a polynomial of at most first degree. Throughout the paper, we demonstrate the wide applicability of our results by showing that many classes of simple-form pairs (V, OMEGA) can be excluded from being VF's.