REVIVING SOME GEOMETRIC ASPECTS OF SHRINKAGE ESTIMATION IN LINEAR MODELS

Abstract

It is well known that the least squares estimator is the best linearunbiased estimator of the parameter vector in a classical linear model. But,it is ëtoo longí as a vector and unreliable, conÖdence intervals are broad forsome components especially in the case of multicollinearity. Shrinkage (contraction) type estimators are e¢ cient remedial tools in order to solve problemscaused by multicollinearity. In this study, we consider a class of componentwiseshrunken estimators with typical members: Mayer and Willkeís contractionestimator, Marquardtís principal component estimator, Hoerl and Kennardísridge estimator, Liuís linear uniÖed estimator and a discrete shrunken estimator. All estimators considered are ìshorterî than the least squares estimatorwith respect to the Euclidean norm, biased, but insensitive to multicollinearityand admissible within the set of linear estimators with respect to unweightedsquared error risk. Some behaviors of these estimators are illustrated geometrically by tracing their tra jectories as functions of shrinkage factors in a twodimensional parameter space.