Integral Inequality for Minimaxity in the Stein Problem

In the estimation of a multivariate normal mean, it is shown that the problem of deriving shrinkage estimators improving on the maximum likelihood estimator can be reduced to that of solving an integral inequality. The integral inequality not only provides a more general condition than a conventional differential inequality studied in the literature, but also handles non-differentiable or discontinuous estimators. The paper also gives general conditions on prior distributions such that the resulting generalized Bayes estimators are minimax. Finally, a simple proof for constructing a class of estimators improving on the James-Stein estimator is given based on the integral expression of the risk.