Robust ridge M-estimators with pretest and Stein-rule shrinkage for an intercept term

If the data contain both multicollinearity and outliers, the ridge M-estimator is the preferred estimator to the usual least square estimator (Silvapulle, Aust J Stat 33:319-333, 1991). Many other estimators, such as the pretest ridge M-estimator and Stein-rule shrinkage ridge M-estimator, have been developed on the basis of the ridge M-estimator. However, all these existing estimators do not consider shrinkage estimation for the intercept term. Hence, there are some rooms for improving the existing estimators by improving the estimator for the intercept term. In this paper, we propose several new ridge M-estimators for regression coefficients and an intercept term by introducing pretest and Stein-rule shrinkage schemes. Our estimators are obtained by using the Jimichi-type ridge matrix that allows shrinkage operations to be applicable to both the intercept term and regression coefficients. We conduct Monte Carlo simulation studies to examine the performance of the proposed estimators. For demonstration, we analyze the corporate finance data from the Nikkei Economic Electronic Databank System in Japan, and the gene expression data from Japanese ovarian cancer patients.