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.

New Facts in Regression Estimation under Conditions of Multicollinearity

“…As for the James-Stein estimator, the benefit is small as shown above. Furthermore, as shown in [18], efficiency of this estimator decreases with increasing 2 R .…”

“…[6]) that excluding components corresponding to small eigenvalues may sharply reduce the quality of the estimation and prediction. In particular, in the brief review of opportunities of the shrinkage estimators made in [18] this fact is confirmed by an analysis of the simulation outcomes in [20]. Finally, as to the James-Stein estimator, in our problem, we cannot apply this method in a canonical form [17], as the number of predictor variables is less than three.…”

“…But in similar circumstances, we must add to the above four types of shrinkage estimators two additional approaches which may be beneficial. We are talking about the Inequality Constrained Least Squares method (ICLS), and the Dual estimator (DE) proposed by the author of this paper in [18].…”

“…Finally, the fifth chapter discusses alternative approaches. These approaches consist in an application of the known Inequality Constrained Least Squares method and the Dual estimator method proposed by the author in [18]. We show that in the presence of a priori information these approaches have significant advantages.…”

“…Let us give some properties of the probability under study. 1) As can be seen from (16)(17)(18)(19)(20)(21) the central moments are even functions of z β ′ . Thus the probability is also an even function.…”