A Characterization of Certain Statistics in Exponential Models Whose Distributions Depend on a Sub-Vector of Parameters Only

Abstract

Let $W$ be a $(k + r)$-dimensional random (column) vector with distribution $F^W_\xi$ belonging to a $(k + r)$-parameter exponential family, $\Pi = \{F^W_\xi: \xi \in \Omega \subset R^{k+r}\}$. Let $\mu$ be a $\sigma$-finite measure which dominates $\Pi$ such that for all $\xi \in \Omega, F^W_\xi$ has a density, with respect to $\mu$, of the form $f^W(w:\xi) = h(w) \exp\{\xi'w + c(\xi)\}$. Consider the partitions $W' = (U', T')$ and $\xi' = (\theta', \nu')$, where $U, \theta \in R^k$ and $T, \nu \in R^r$. It is proven that the conditional covariance matrix of $U$ given $T = t$ does not depend on $t$ for almost all values of $t$ if and only if there exists a unique measurable vector-valued function $g(T) = (g_1(T), \cdots, g_k(T))'$, such that the random vector $Z = U - g(T)$ is stochastically independent of $T$ under any member of $\Pi$. Furthermore, the distribution of $Z$ is shown to constitute a $k$-parameter exponential family with $\theta$ as the vector of natural parameters. Further results are obtained and exemplified.