Bates and best quadratic unbiased estimators for parameters of the covariance matrix in a normal linear model

Let y = X β + ε be a linear model with ε normally distributed such that Eε = 0, . Starting from prior distributions with respect to β and σ the authors develope the conception of BAQUE (BAYEsian Quadratic Unbiased Estimator) for estimation of linear functions of the σ i. It is shown that BAQUE depends only on the first and second moments of the prior distributions, and a method for its computation is given. Conditions for the independence of BAQUE of prior distributions are derived. i.e. conditions for the existence of uniformly best quadratic unbiased estimators. Further, BAQUE is used for the existence of uniformly best quadratic unbiased estimators. Further, BAQUE is used for estimation in the variance components model, and is compared with MINQUE (introduced by C.R. RAO). MINQUE is BAQUE for special prior distributions. Finally applications to multivariate regression model and to other models are given.