Reproductive Exponential Families

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