A new class of generalized Bayes minimax ridge regression estimators

Abstract

Let y=Aβ+ε, where y is an N\times1 vector of observations, βis a p\times1 vector of unknown regression coefficients, A is an N\times p design matrix and εis a spherically symmetric error term with unknown scale parameter σ. We consider estimation of βunder general quadratic loss functions, and, in particular, extend the work of Strawderman [J. Amer. Statist. Assoc. 73 (1978) 623-627] and Casella [Ann. Statist. 8 (1980) 1036-1056, J. Amer. Statist. Assoc. 80 (1985) 753-758] by finding adaptive minimax estimators (which are, under the normality assumption, also generalized Bayes) of β, which have greater numerical stability (i.e., smaller condition number) than the usual least squares estimator. In particular, we give a subclass of such estimators which, surprisingly, has a very simple form. We also show that under certain conditions the generalized Bayes minimax estimators in the normal case are also generalized Bayes and minimax in the general case of spherically symmetric errors.