Reviving some geometric aspects of shrinkage estimation in linear models

Abstract

It is well known that the least squares estimator is the best linear unbiased estimator of the parameter vector in a classical linear model. But, it is `too long' as a vector and unreliable, confidence intervals are broad for some components especially in the case of multicollinearity. Shrinkage (contraction) type estimators are efficient remedial tools in order to solve problems caused by multicollinearity. In this study, we consider a class of componentwise shrunken estimators with typical members: Mayer and Willke's contraction estimator, Marquardt's principal component estimator, Hoerl and Kennard's ridge estimator, Liu's linear unified estimator and a discrete shrunken estimator. All estimators considered are shorter than the least squares estimator with respect to the Euclidean norm, biased, but insensitive to multicollinearity and admissible within the set of linear estimators with respect to unweighted squared error risk. Some behaviors of these estimators are illustrated geometrically by tracing their trajectories as functions of shrinkage factors in a two- dimensional parameter space.