January 2004
We consider frequentist and empirical Bayes estimation of linear regression coefficients with T observations and K orthonormal regressors. The frequentist formulation considers estimators that are equivariant under permutations of the regressors. The empirical Bayes formulation (both parametric and nonparametric) treats the coefficients as i.i.d. and estimates their prior. Asymptotically; when K = ρTδ for 0 < ρ < 1 and 0 < δ ≤ 1; the empirical Bayes estimator is shown to be: (i) optimal in Robbins’ (1955; 1964) sense; (ii) the minimum risk equivariant estimator; and (iii) minimax in both the frequentist and Bayesian problems over a wide class of error distributions. Also; the asymptotic frequentist risk of the minimum risk equivariant estimator is shown to equal the Bayes risk of the (infeasible subjectivist) Bayes estimator in the Gaussian model with a “prior” that is the weak limit of the empirical c.d.f. of the true parameter values.