Improved estimation of a patterned covariance matrix

Suppose a random vector X has a multinormal distribution with covariance matrix [Sigma] of the form [Sigma] = [Sigma]i=1k [theta]iMi, where Mi's form a known complete orthogonal set and [theta]i's are the distinct unknown eigenvalues of [Sigma]. The problem of estimation of [Sigma] is considered under several plausible loss functions. The approach is to establish a duality relationship: estimation of the patterned covariance matrix [Sigma] is dual to simulataneous estimation of scale parameters of independent [chi]2 distributions. This duality allows simple estimators which, for example, improve upon the MLE of [Sigma]. It also allows improved estimation of tr [Sigma]. Examples are given in the case when [Sigma] has equicorrelated structure.