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September, 1976 Characterization of Prior Distributions and Solution to a Compound Decision Problem
C. Radhakrishna Rao
Ann. Statist. 4(5): 823-835 (September, 1976). DOI: 10.1214/aos/1176343583


Let $y = \theta + e$ where $\theta$ and $e$ are independent random variables so that the regression of $y$ on $\theta$ is linear and the conditional distribution of $y$ given $\theta$ is homoscedastic. We find prior distributions of $\theta$ which induce a linear regression of $\theta$ on $y$. If in addition, the conditional distribution of $\theta$ given $y$ is homoscedastic (or weakly so), then $\theta$ has a normal distribution. The result is generalized to the Gauss-Markoff model $\mathbf{Y} = \mathbf{X\theta} + \mathbf{\varepsilon}$ where $\mathbf{\theta}$ and $\mathbf{\varepsilon}$ are independent vector random variables. Suppose $\bar{y}_i$ is the average of $p$ observations drawn from the $i$th normal population with mean $\theta_i$ and variance $\sigma_0^2$ for $i = 1,\cdots, k$, and the problem is the simultaneous estimation of $\theta_1,\cdots, \theta_k$. An estimator alternative to that of James and Stein is obtained and shown to have some advantage.


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C. Radhakrishna Rao. "Characterization of Prior Distributions and Solution to a Compound Decision Problem." Ann. Statist. 4 (5) 823 - 835, September, 1976.


Published: September, 1976
First available in Project Euclid: 12 April 2007

zbMATH: 0341.62029
MathSciNet: MR426234
Digital Object Identifier: 10.1214/aos/1176343583

Primary: 62C10
Secondary: 62C25

Keywords: characterization problems , compound decision , Empirical Bayes , Linear regression , prior distribution , simultaneous estimation

Rights: Copyright © 1976 Institute of Mathematical Statistics

Vol.4 • No. 5 • September, 1976
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