Improved estimation of a patterned covariance matrix

AbstractSuppose a random vector X has a multinormal distribution with covariance matrix Σ of the form Σ = Σi=1k θiMi, where Mi's form a known complete orthogonal set and θi's are the distinct unknown eigenvalues of Σ. The problem of estimation of Σ is considered under several plausible loss functions. The approach is to establish a duality relationship: estimation of the patterned covariance matrix Σ is dual to simulataneous estimation of scale parameters of independent χ2 distributions. This duality allows simple estimators which, for example, improve upon the MLE of Σ. It also allows improved estimation of tr Σ. Examples are given in the case when Σ has equicorrelated structure