Shrinkage estimation with singular priors and an application to small area estimation

Abstract

The paper considers estimation of the multivariate normal mean under a multivariate normal prior with a singular precision matrix. Such a setup appears in the multi-task averaging, serial and spatial smoothing problems. The empirical and hierarchical Bayes estimators shrink the maximum likelihood estimator by projecting it to the null space of the precision matrix. Conditions for minimaxity are given for the estimators proposed in this paper. The singular prior is applied to the Fay–Herriot small area estimation model with random effects having the singular distribution. Second-order approximation of the conditional mean squared error of the empirical Bayes estimator and its second-order unbiased estimator are derived. Numerical simulations confirm that the derived estimators perform well under the situation when there is a spatial correlation in the sample.