Contracting towards subspaces when estimating the mean of a multivariate normal distribution

Abstract

The problem of estimating, under unweighted quadratic loss, the mean of a multinormal random vector X with arbitrary covariance matrix V is considered. The results of James and Stein for the case V = I have since been extended by Bock to cover arbitrary V and also to allow for contracting X towards a subspace other than the origin; minimax estimators (other than X) exist if and only if the eigenvalues of V are not “too spread out.” In this paper a slight variation of Bock's estimator is considered. A necessary and sufficient condition for the minimaxity of the present estimator is (∗): the eigenvalues of ( I − P) V should not be “too spread out,” where P denotes the projection matrix associated with the subspace towards which X is contracted. The validity of (∗) is then examined for a number of patterned covariance matrices (e.g., intraclass covariance, tridiagonal and first order autocovariance) and conditions are given for (∗) to hold when contraction is towards the origin or towards the common mean of the components of X. (∗) is also examined when X is the usual estimate of the regression vector in multiple linear regression. In several of the cases considered the eigenvalues of V are “too spread out” while those of ( I − P) V are not, so that in these instances the present method can be used to produce a minimax estimate.