A Dual estimator as a tool for solving regression problems

Abstract

This paper discusses a parameter estimation method that employs an unusual estimator called the Dual estimator. For a linear regression model, we obtain two alternative estimators by subtracting or adding a certain vector to the vector of the Ordinary Least Squares Estimator (OLSE). One of them strictly dominates the latter. Moreover, under the normality assumption this estimator is unbiased, and consistent, and has significantly smaller variance than the OLSE. The use of a priori information is a universal way to choose a better alternative. An important property of the proposed method is the possibility of using the strict inequalities as a priori information. In particular, if the external information is that the L2-norm of the OLS estimate exceeds the same norm of a vector of true coefficients, one can choose a better alternative without additional parameters. If it is known that the parameter is restricted by a linear non-strict inequality, the method has a smaller Mean Squared Error than a Constrained Least Squares technique. Finally, a priori information on two possible parameter values can be successfully used for the experimental confirmation of one of two alternative theories, which is illustrated by a verification of the General Theory of Relativity based upon astronomical data.