Anomalies in the Foundations of Ridge Regression

Abstract

Anomalies persist in the foundations of ridge regression as set forth in Hoerl and Kennard (1970) and subsequently. Conventional ridge estimators and their properties do not follow on constraining lengths of solution vectors using LaGrange's method, as claimed. Estimators so constrained have singular distributions; the proposed solutions are not necessarily minimizing; and heretofore undiscovered bounds are exhibited for the ridge parameter. None of the considerable literature on estimation, prediction, cross--validation, choice of ridge parameter, and related issues, collectively known as ridge regression, is consistent with constrained optimization, nor with corresponding inequality constraints. The problem is traced to a misapplication of LaGrange's principle, failure to recognize the singularity of distributions, and misplaced links between constraints and the ridge parameter. Other principles, based on condition numbers, are seen to validate both conventional ridge and surrogate ridge regression to be defined. Numerical studies illustrate that ridge analysis often exhibits some of the same pathologies it is intended to redress.