Best equivariant estimation in curved covariance models

AbstractLet X1, …, Xn (n > p > 2) be independently and identically distributed p-dimensional normal random vectors with mean vector μ and positive definite covariance matrix Σ and let Σ and S = Σα = 1n(Xα − X)(Xα − X)′. X = (1n) Σα = 1n Xα be partioned as1 p−1 1 p−1∑=∑11∑12∑21∑221p−1,S=S11S12S21S221p−1. We derive here the best equivariant estimators of the regression coefficient vector β = Σ22−1Σ21 and the covariance matrix Σ22 of covariates given the value of the multiple correlation coefficient ϱ2 = Σ11−1Σ12Σ22−1Σ21. Such problems arise in practice when it is known that ϱ2 is significant. Let R2 = S11−1S12S22−1S21. If the value of ϱ2 is such that terms of order (Rϱ)2 and higher can be neglected, the best equivariant estimator of β is approximately equal to (n −1)(p − 1)−1 ϱ2S22−1S21, where S22−1S21 is the maximum likelihood estimator of β. When ϱ2 = 0, the best equivariant estimator of Σ22 is (n − p + 1)−1S22 is the maximum likelihood estimator of Σ22