Best equivariant estimation in curved covariance models

Let X1, ..., Xn (n > p > 2) be independently and identically distributed p-dimensional normal random vectors with mean vector [mu] and positive definite covariance matrix [Sigma] and let [Sigma] and . be partioned as1 p-1 1 p-1. We derive here the best equivariant estimators of the regression coefficient vector [beta] = [Sigma]22-1[Sigma]21 and the covariance matrix [Sigma]22 of covariates given the value of the multiple correlation coefficient [varrho]2 = [Sigma]11-1[Sigma]12[Sigma]22-1[Sigma]21. Such problems arise in practice when it is known that [varrho]2 is significant. Let R2 = S11-1S12S22-1S21. If the value of [varrho]2 is such that terms of order (R[varrho])2 and higher can be neglected, the best equivariant estimator of [beta] is approximately equal to (n -1)(p - 1)-1 [varrho]2S22-1S21, where S22-1S21 is the maximum likelihood estimator of [beta]. When [varrho]2 = 0, the best equivariant estimator of [Sigma]22 is (n - p + 1)-1S22 is the maximum likelihood estimator of [Sigma]22.