The Annals of Statistics

Reproductive Exponential Families

O. Barndorff-Nielsen and P. Blaesild

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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

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

First available in Project Euclid: 12 April 2007

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


Primary: 62E15: Exact distribution theory
Secondary: 62F99: None of the above, but in this section

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


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

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