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Variance functions with meromorphic means. (English) Zbl 0735.62010

A natural exponential family is characterized by a pair \((\Omega,V)\) where \(\Omega\), the mean domain, is an open interval in \(\mathbb{R}\) and \(V\) is the associated variance function, regarded as a function of the mean. The authors make a further contribution to the problem of characterizing the set of possible pairs \((\Omega,V)\), a question of interest for the construction of generalized linear models.
One of their main results states that, if the mean function has a meromorphic continuation to \(\mathbb{C}\) and if, further, \(V\) has a unique analytic continuation to \(\mathbb{C}\), except for isolated singularities, then \(V\) must be a polynomial of degree at most two. All proofs are based on complex analysis methods.
Reviewer: R.Grübel

MSC:

62E10 Characterization and structure theory of statistical distributions
30E99 Miscellaneous topics of analysis in the complex plane
30B40 Analytic continuation of functions of one complex variable
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