## The Annals of Statistics

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

### Natural Real Exponential Families with Cubic Variance Functions

Gerard Letac and Marianne Mora

#### Abstract

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

**Source**

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

**Dates**

First available in Project Euclid: 12 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aos/1176347491

**Digital Object Identifier**

doi:10.1214/aos/1176347491

**Mathematical Reviews number (MathSciNet)**

MR1041384

**Zentralblatt MATH identifier**

0714.62010

**JSTOR**

links.jstor.org

**Subjects**

Primary: 62E10: Characterization and structure theory

Secondary: 60J30

**Keywords**

Natural exponential families variance functions

#### Citation

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. https://projecteuclid.org/euclid.aos/1176347491