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.
Citation
H. M. Hudson. "A Natural Identity for Exponential Families with Applications in Multiparameter Estimation." Ann. Statist. 6 (3) 473 - 484, May, 1978. https://doi.org/10.1214/aos/1176344194
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