Characterization of Prior Distributions and Solution to a Compound Decision Problem

The Annals of Statistics · 1976 · Volume 4 · Issue 5

Abstract

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.

Keywords

62C10, 62C25, characterization problems, compound decision, Empirical Bayes, Linear regression, prior distribution, simultaneous estimation

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