Hypercube estimators: Penalized least squares, submodel selection, and numerical stability

Abstract

Hypercube estimators for the mean vector in a general linear model include algebraic equivalents to penalized least squares estimators with quadratic penalties and to submodel least squares estimators. Penalized least squares estimators necessarily break down numerically for certain penalty matrices. Equivalent hypercube estimators resist this source of numerical instability. Under conditions, adaptation over a class of candidate hypercube estimators, so as to minimize the estimated quadratic risk, also minimizes the asymptotic risk under the general linear model. Numerical stability of hypercube estimators assists trustworthy adaptation. Hypercube estimators have broad applicability to any statistical methodology that involves penalized least squares. Notably, they extend to general designs the risk reduction achieved by Stein’s multiple shrinkage estimators for balanced observations on an array of means.