In this paper, we consider the estimation problem of the weighted least absolute deviation (WLAD) regression parameter vector when there are some outliers or heavy-tailed errors in the response and the leverage points in the predictors. We propose the pretest and James–Stein shrinkage WLAD estimators when some of the parameters may be subject to certain restrictions. We derive the asymptotic risk of the pretest and shrinkage WLAD estimators and show that if the shrinkage dimension exceeds two, the asymptotic risk of the shrinkage WLAD estimator is strictly less than the unrestricted WLAD estimator. On the other hand, the risk of the pretest WLAD estimator depends on the validity of the restrictions on the parameters. Furthermore, we study the WLAD absolute shrinkage and selection operator (WLAD-LASSO) and compare its relative performance with the pretest and shrinkage WLAD estimators. A simulation study is conducted to evaluate the performance of the proposed estimators relative to that of the unrestricted WLAD estimator. A real-life data example using body fat study is used to illustrate the performance of the suggested estimators.

The ridge regression estimator is a commonly used procedure to deal with multicollinear data. This paper proposes an estimation procedure for high-dimensional multicollinear data that can be alternatively used. This usage gives a continuous estimate, including the ridge estimator as a particular case. We study its asymptotic performance for the growing dimension, i.e., $p\to \infty$ when n is fixed. Under some mild regularity conditions, we prove the proposed estimator’s consistency and derive its asymptotic properties. Some Monte Carlo simulation experiments are executed in their performance, and the implementation is considered to analyze a high-dimensional genetic dataset.