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March, 1990 Natural Real Exponential Families with Cubic Variance Functions
Gerard Letac, Marianne Mora
Ann. Statist. 18(1): 1-37 (March, 1990). DOI: 10.1214/aos/1176347491


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.


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Gerard Letac. Marianne Mora. "Natural Real Exponential Families with Cubic Variance Functions." Ann. Statist. 18 (1) 1 - 37, March, 1990.


Published: March, 1990
First available in Project Euclid: 12 April 2007

zbMATH: 0714.62010
MathSciNet: MR1041384
Digital Object Identifier: 10.1214/aos/1176347491

Primary: 62E10
Secondary: 60J30

Rights: Copyright © 1990 Institute of Mathematical Statistics


Vol.18 • No. 1 • March, 1990
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