On the mean square error of parameter estimates for some biased estimators

Biased estimators of /spl beta/ in the linear statistical model y= ?/spl beta/ + e have been studied by several scholars. Many solutions are of the form /spl beta//spl I.cap/* = (C + X' X)/sup +/X'Y for some compatible matrix C. In this note, conditions on the matrix C are developed which show when, component by component, the mean square error of the biased estimator is less than the corresponding mean square error, or variance, of the estimators using the normal equations. That is if /spl beta//spl I.cap/* is the ith component of /spl beta//spl I.cap/*, and /spl beta//spl I.cap//sub i/ is the ith component of /spl beta//spl I.cap/ = (C + ?'?)/sup -/ X'Y, C is determined so that for each i var[/spl beta//spl I.cap/*] + (/spl beta//sub i/* - (/spl beta//sub i/)/sup 2/]