A Natural Identity for Exponential Families with Applications in Multiparameter Estimation

Abstract

A random variable $X$ is said to have distribution in the class $\mathscr{E}_0$ if, for some real valued, positive function $a(\bullet)$, the identity $E\{(X - \mu)g(X)\} = E\{a(X)g'(X)\}$ holds for any absolutely continuous real valued function $g(\bullet)$ satisfying $E|a(X)g'(X)| < \infty$. Examples of a random variable $X$ possessing a distribution in $\mathscr{E}_0$ are (i) $X$ normally distributed with mean $\mu$ and standard deviation 1, (ii) $X$ having a gamma density with mean $\mu$ and location parameter 1, (iii) $X = 1/Y$ where $Y \sim \lbrack(n - 2)\rbrack^{-1}\chi_n^2, n > 2$. Suppose $X_1,\cdots, X_p, p \geqq 3$, are independently distributed with distributions in $\mathscr{E}_0$, for some function $a(\bullet)$, and with means $\mu_1,\cdots, \mu_p$. Define $b(x) = \int a(x)^{-1} dx$, where the integral is interpreted as indefinite, $B_i = b(X_i), S = \sum^p_{i=1} B_i^2, X' = (X_1,\cdots, X_p)$ and $B' = (B_1,\cdots, B_p)$. Then the estimator $X - ((p - 2)/S)B$ dominates $X$ if sum of squared error loss is assumed. Similar estimators are obtained, when $p \geqq 4$, which shrink towards an origin determined by the data. There are corresponding results for some discrete exponential families.