Characterization of Prior Distributions and Solution to a Compound Decision Problem

Let y = θ + e where θ and e are independent random variables so that the regression of y on θ is linear and the conditional distribution of y given θ is homoscedastic. We find prior distributions of θ which induce a linear regression of θ on y. If in addition, the conditional distribution of θ given y is homoscedastic (or weakly so), then θ has a normal distribution. The result is generalized to the Gauss-Markoff model Y = Xθ + ε where θ and ε are independent vector random variables. Suppose ȳᵢ is the average of p observations drawn from the ith normal population with mean θᵢ and variance σ₀² for i = 1,⋯, k, and the problem is the simultaneous estimation of θ₁,⋯, θₖ. An estimator alternative to that of James and Stein is obtained and shown to have some advantage.