Bar-Lev, Shaul K.; Bshouty, Daoud; Enis, Peter Variance functions with meromorphic means. (English) Zbl 0735.62010 Ann. Probab. 19, No. 3, 1349-1366 (1991). 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 Cited in 7 Documents 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 Keywords:exponential dispersion model; meromorphic mean functions; Laplace transform; meromorphic variance function; reciprocity; natural exponential family; variance function; construction of generalized linear models; meromorphic continuation; analytic continuation PDFBibTeX XMLCite \textit{S. K. Bar-Lev} et al., Ann. Probab. 19, No. 3, 1349--1366 (1991; Zbl 0735.62010) Full Text: DOI