Some Liu and ridge-type estimators and their properties under the ill-conditioned Gaussian linear regression model

The estimation of the regression parameters for the ill-conditioned Gaussian linear regression model are considered in this paper. Accordingly, we consider some improved Liu [A new class of biased estimate in linear regression, Commun. Stat. Theory Methods 22 (1993), pp. 393-402] type estimators, namely the unrestricted Liu estimator, restricted Liu estimator and the preliminary test Liu estimator (PTLE) for estimating the regression parameters. The performances of the proposed estimators are compared based on the quadratic bias and risk functions under both null and alternative hypotheses. The conditions of superiority of the proposed estimators for departure parameter, Delta, and biasing parameter, d, are given. We also numerically compared the performance of PTLE with the preliminary test ridge regression estimator (PTRRE) and concluded that for small values of d and ridge parameter k, PTLE performed better than the PTRRE; otherwise the PTRRE performed better than PTLE in the sense of smaller MSE.