Multivariate weighted least squares as a preferable alternative to the determinant criterion for multiresponse parameter estimation

Abstract

Box and Draper's (1965) determinant criterian for multiresponse parameter estimation is commonly used in preference to ordinary least squares when the measurement error covariance matrix is unknown. Phillips (1976) has shown that the determinant criterion is numerically equivalent to an iterated generalized least squares scheme. From this equivalence, it is shown that, of all such weighting schemes, the determinant criterion in a certain sense minimizes the estimated parameter variances. However, when the number of sets of measurements is not large relative to the number of responses, Monte-Carlo simulation reveals that a multivariate weighted least squares scheme can give parameter variances that are smaller than those given by the determinant criterion. This suggests that the optimality property of the determinant criterion cited above is only asymptotically valid. Monte-Carlo simulation also reveals that, in contrast to multivariate weighted least squares, the determinant criterion can yield parameter estimates whose frequency distribution is very far from normal in the tails. Multivariate weighted least squares (MWLS) is therefore recommended as a robust alternative to the determinant criterion for multiresponse parameter estimation.