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

Biased estimators of β in the linear statistical model y= ?β + e have been studied by several scholars. Many solutions are of the form β&capped;* = (C + X´ X)⁺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 β&capped;* is the ith component of β&capped;*, and β&capped;_i is the ith component of β&capped; = (C + ?´?)^- X´Y, C is determined so that for each i var[β&capped;*] + (β_i* - (β_i)²] i]. The implication for Hoerl and Kennard´s Marquardt´s and Mayer and Wilke´s biased estimators are discussed. Subsequently, the singular valued decompositions of X are used to explore β&capped; - β&capped; *.