## The Annals of Statistics

- Ann. Statist.
- Volume 11, Number 3 (1983), 770-782.

### Reproductive Exponential Families

O. Barndorff-Nielsen and P. Blaesild

#### Abstract

Consider a full and steep exponential model $\mathscr{M}$ with model function $a(\theta)b(x)\exp\{\theta \cdot t(x)\}$ and a sample $x_1, \cdots, x_n$ from $\mathscr{M}$. Let $\bar{t} = \{t(x_1) + \cdots + t(x_n)\}/n$ and let $\bar{t} = (\bar{t}_1, \bar{t}_2)$ be a partition of the canonical statistic $\bar{t}$. We say that $\mathscr{M}$ is reproductive in $t_2$ if there exists a function $H$ independent of $n$ such that for every $n$ the marginal model for $\bar{t}_2$ is exponential with $n\theta$ as canonical parameter and $(H(\bar{t}_2), \bar{t}_2)$ as canonical statistic. Furthermore we call $\mathscr{M}$ strongly reproductive if these marginal models are all contained in that for $n = 1$. Conditions for these properties to hold are discussed. Reproductive exponential models are shown to allow of a decomposition theorem analogous to the standard decomposition theorem for $\chi^2$-distributed quadratic forms in normal variates. A number of new exponential models are adduced that illustrate the concepts and also seem of some independent interest. In particular, a combination of the inverse Gaussian distributions and the Gaussian distributions is discussed in detail.

#### Article information

**Source**

Ann. Statist., Volume 11, Number 3 (1983), 770-782.

**Dates**

First available in Project Euclid: 12 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aos/1176346244

**Digital Object Identifier**

doi:10.1214/aos/1176346244

**Mathematical Reviews number (MathSciNet)**

MR707928

**Zentralblatt MATH identifier**

0529.62016

**JSTOR**

links.jstor.org

**Subjects**

Primary: 62E15: Exact distribution theory

Secondary: 62F99: None of the above, but in this section

**Keywords**

Affine foliations decomposition exact tests generalized linear models independence inverse Gaussian distribution

#### Citation

Barndorff-Nielsen, O.; Blaesild, P. Reproductive Exponential Families. Ann. Statist. 11 (1983), no. 3, 770--782. doi:10.1214/aos/1176346244. https://projecteuclid.org/euclid.aos/1176346244