Bayes Estimators of Covariance Parameters and the Influence of Designs

Abstract

It is assumed that the covariance matrix of N observations has the form $C_\theta = \sum\nolimits_{r = 1}^R {\theta _r U_r }$ where U ₁,...,U _R are known covariance matrices and θ ₁,...,θ _R are unknown parameters. Estimators for $\sum\nolimits_{r = 1}^R {\theta _r b_r }$ with known b ₁,...,b _R are characterized which minimize the Bayes risk within all invariant quadratic unbiased estimators. In this characterization, the matrix A, which determines the quadratic form of the estimator, is given by a linear equation system which is not of full rank. It is shown that some solutions of the equation system prove to be asymmetric matrices A. Therefore, sufficient conditions are presented which ensures symmetry of the matrix A. Given this result, the influence of designs on the Bayes risk is studied.