The Annals of Statistics

Natural Real Exponential Families with Cubic Variance Functions

Gerard Letac and Marianne Mora

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Pursuing the classification initiated by Morris (1982), we describe all the natural exponential families on the real line such that the variance is a polynomial function of the mean with degree less than or equal to 3. We get twelve different types; the first six appear in the fundamental paper by Morris (1982); most of the other six appear as distributions of first passage times in the literature, the inverse Gaussian type being the most famous example. An explanation of this occurrence of stopping times is provided by the introduction of the notion of reciprocity between two measures or between two natural exponential families, and by classical fluctuation theory.

Article information

Ann. Statist., Volume 18, Number 1 (1990), 1-37.

First available in Project Euclid: 12 April 2007

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Zentralblatt MATH identifier


Primary: 62E10: Characterization and structure theory
Secondary: 60J30

Natural exponential families variance functions


Letac, Gerard; Mora, Marianne. Natural Real Exponential Families with Cubic Variance Functions. Ann. Statist. 18 (1990), no. 1, 1--37. doi:10.1214/aos/1176347491.

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