On Biased Estimation in Linear Models

Hoer1 and Kennard introduced a class of biased estimators (ridge estimators) for the parameters in an ill-conditioned linear model. In this paper the ridge estimators are viewed as a subclass of the class of linear transforms of the least squares estimator. An alternative class of estimators, labeled shrunken estimators is considered. It is shown that these estimators satisfy the admissibility condition proposed by Hoer1 and Kennard. In addition, both the ridge estimators and shrunken estimators are derived as minimum norm estimators in the class of linear transforms of the least squares estimators. The former minimizes the Euclidean norm and the latter minimizes the design dependent norm. The class of estimators which are minimum variance linear transforms of the least squares estimator is obtained and the members of this class are shown to be stochastically shrunken estimators. An example is computed to show the behavior of the different estimators.