Estimating the Mean of a Normal Distribution with Loss Equal to Squared Error Plus Complexity Cost

Abstract

Estimating the mean of a $p$-variate normal distribution is considered when the loss is squared error plus a complexity cost. The complexity of estimates is defined using a partition of the parameter space into sets corresponding to models of different complexity. The model implied by the use of an estimate determines the estimate's complexity cost. Complete classes of estimators are developed which consist of preliminary-test estimators. As is the case when loss is just squared error, the maximum-likelihood estimator is minimax. However, unlike the no-complexity-cost case, the maximum-likelihood estimator is inadmissible even in the case when $p = 1$ or 2.