Anomalies in the Foundations of Ridge Regression: Some Clarifications

Several anomalies in the foundations of ridge regression from the perspective of constrained least-square (LS) problems were pointed out in Jensen & Ramirez. Some of these so-called anomalies, attributed to the non-monotonic behaviour of the norm of unconstrained ridge estimators and the consequent lack of sufficiency of Lagrange's principle, are shown to be incorrect. It is noted in this paper that, for a fixed <formula format="inline"><simplemath>Y</simplemath></formula>, norms of unconstrained ridge estimators corresponding to the given basis are indeed strictly monotone. Furthermore, the conditions for sufficiency of Lagrange's principle are valid for a suitable range of the constraint parameter. The discrepancy arose in the context of one data set due to confusion between estimates of the parameter vector, <formula format="inline"><simplemath>β</simplemath></formula>, corresponding to different parametrization (choice of bases) and/or constraint norms. In order to avoid such confusion, it is suggested that the parameter <formula format="inline"><simplemath>β</simplemath></formula> correspondi ng to each basis be labelled appropriately. Copyright (c) 2010 The Authors. Journal compilation (c) 2010 International Statistical Institute.