Hypercube Estimators: Penalized Least Squares, Submodel Selection and
Numerical Stability

Rudolf Beran
University of California, Davis


Classes of penalized least squares (PLS) estimators with quadratic
penalty terms generally have, as limit points, certain least squares
submodel fits. Expressed in their customary form, as generalized ridge
estimators, the PLS fits are highly ill-conditioned near these limit
submodel fits. Hypercube estimators embed PLS estimators and their
limit submodel fits in an algebraic form that enables smooth
interpolation amongst them. Unlike their PLS parents, hypercube
estimators have bounded condition number. Numerical stability enables
identifying  hypercube estimators that minimize estimated risk and,
thereby, asymptotic risk.  Applied to fitting an array of means
observed with error, risk-adaptive hypercube estimators extend to
unbalanced designs the risk-reduction and interpretability achieved by
multiple Stein shrinkage in balanced designs.