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

## 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

## Information

Published: May, 1978
First available in Project Euclid: 12 April 2007

zbMATH: 0391.62006
MathSciNet: MR467991
Digital Object Identifier: 10.1214/aos/1176344194

Subjects:
Primary: 62C15
Secondary: 62C25 , 62F10

Keywords: empirical Bayes estimation , James-Stein estimator , multiparameter estimation , squared error loss  