A note on the comparison of minimax linear and mixed regression estimation of regression coefficients when prior estimates are available

When prior estimates of regression coe cients along with their stan dard errors or their variance covariance matrix are available they can be incorporated into the estimation procedure through minimax linear and mixed regression approaches It is demonstrated that the mixed regres sion approach provides more e cient estimators at least asymptotically in comparison to the minimax linear approach with respect to the criterion of variance covariance matrix keywords linear regression mixed model Introduction Recent studies past experience and pilot investigations often provide some use ful information in the form of estimates of coe cients in a linear regression model along with their standard errors or their variance covariance matrix Such a prior information can be incorporated into the estimation procedure in two simple ways based on point and interval estimation of parameters One is to express it as a set of stochastic linear restrictions and to apply the method of mixed regression estimation And the other is to form a con dence ellipsoid for the regression coe cients and to apply the method of minimax linear esti mation The latter approach generally does not lead to a simple form of the estimators and iterative procedures are to be followed see e g Schipp for an interesting exposition However if we take the matrix involved in the quadratic loss function to be of rank one the estimators have a closed form see e g Rao and Toutenburg Chap Here we restrict our attention to such estimators Superiority of estimators arising from both the mixed regression and mini max linear estimation procedures over the conventional estimators ignoring the prior information is well discussed in the literature but explicit attention does not seem to have been paid to the relative e ciency of one approach over the other see e g Rao and Toutenburg for the details This has formed the subject matter of this note Taking the performance criterion to be the variance covariance matrix it is found that the mixed regression approach provides more e cient estimators than the minimax linear approach at least asymptotically Main Result Consider a linear regression model y X u wehere y is a n vector of n observations on the study variable X is a n K matrix of n observations on K explanatory variables is a K vector of regression coe cients and u is a n vector of disturbances following a multivariate normal distribution with null mean vector as and variance covariance matrix as times an identity matrix It is assumed that the matrix X has full column rank and the scalar is unknown Further let us assume to be given the prior information specifying an unbi ased estimate b along with the variance covariance matrix s X X based on n observations from some extraneous source This prior information can be expressed in two forms for the purpose of utilizing it in the estimation of From the viewpoint of point estimation we may write it as