"Minimax Empirical Bayes Ridge-Principal Component Regression Estimators"

In this paper, we consider the problem of estimating the regression parameters in a multiple linear regression model with design matrix A when the multicollinearity is present. Minimax empirical Bayes estimators are proposed under the assumption of normality and loss function (ƒÂ-s)t (At A)2 (ƒÂ- s)/ƒÐ2, where ƒÂ is an estimator of the vector s of p regression parameters, and ƒÐ2 is the unknown variance of the model. The minimax estimators are also obtained under linear constraints on s such as s = Cƒ¿ for some p ~ q known matrix C, q <_ p. For a particular C, this combines the principal component regression and ridge regression. These results are also applicable for estimating the p means ƒÆi when the p observations xi are independently distributed as N (ƒÆi, diƒÐ2), di's are known butƒÐ2 is unknown.