Performance of Positive Rule Estimator in the Ill-Conditioned Gaussian Regression Model

Ridge regression is a widely used method to estimate the regression parameters for an ill-conditioned model. This paper describes the estimation of the regression parameters for the Gaussian linear regression model with ill-conditioned explanatory variables. We propose some improved estimators, namely, the unrestricted ridge regression estimator, restricted ridge regression estimator, preliminary test ridge regression estimator, shrinkage ridge regression estimator and positive rule ridge regression estimators in this paper. The performances of the proposed estimators are compared based on the quadratic bias and risk functions under both null and alternative hypotheses, which specify certain restrictions on the regression parameters. The conditions of superiority of the proposed estimators for departure and ridge parameters are given. It is demonstrated that unlike the positive rule shrinkage (PR) estimator which dominates both unrestricted and shrinkage estimators, the positive rule ridge regression estimator (PRRRE) utilizes both sample and non-sample information but does not outperform the unrestricted and shrinkage ridge regression estimators for an ill-conditioned data. Some graphical representations have been presented which support the findings of the paper.