THE ROLE OF THE COVARIANCE-MATRIX IN THE LEAST-SQUARES ESTIMATION FORA COMMON-MEAN

For n > I let X = (X-1,..., X-n)' have a mean vector BI and covariance matrix sigma(2) Sigma, where 1=(1,...,1)', Sigma is a known positive definite matrix, and sigma(2) > 0 is either known or unknown. This mod el has been found useful when the observations X-1,...,X-n from a popu lation with mean theta B are not independent. We show how the variance of <(theta)over cap>, the least-squares estimator of theta, depends o n the covariance structure of Sigma. More specifically, we give expres sions for Var(<(theta)over cap>), obtain its lower and upper bounds (w hich involve only the smallest and the largest eigenvalues of Sigma), and show how the dependence of X-1,..., X-n plays a role in Var<(theta )over cap>. Examples of applications are given for M-matrices, for exc hangeable random variables, for a class of covariance matrices with a block-correlation structure, and for twin data. (C) 1997 Elsevier Scie nce Inc.