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Anomalies in the Foundations of Ridge Regression: Some Clarifications

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  • Prasenjit Kapat
  • Prem K. Goel

Abstract

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 Y , 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, β , corresponding to different parametrization (choice of bases) and/or constraint norms. In order to avoid such confusion, it is suggested that the parameter β correspondi ng to each basis be labelled appropriately. Copyright (c) 2010 The Authors. Journal compilation (c) 2010 International Statistical Institute.

Suggested Citation

  • Prasenjit Kapat & Prem K. Goel, 2010. "Anomalies in the Foundations of Ridge Regression: Some Clarifications," International Statistical Review, International Statistical Institute, vol. 78(2), pages 209-215, August.
  • Handle: RePEc:bla:istatr:v:78:y:2010:i:2:p:209-215
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    References listed on IDEAS

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    1. Sundaram,Rangarajan K., 1996. "A First Course in Optimization Theory," Cambridge Books, Cambridge University Press, number 9780521497190, March.
    2. Sundaram,Rangarajan K., 1996. "A First Course in Optimization Theory," Cambridge Books, Cambridge University Press, number 9780521497701, March.
    3. Donald R. Jensen & Donald E. Ramirez, 2008. "Anomalies in the Foundations of Ridge Regression," International Statistical Review, International Statistical Institute, vol. 76(1), pages 89-105, April.
    4. Sylvain Sardy, 2008. "On the Practice of Rescaling Covariates," International Statistical Review, International Statistical Institute, vol. 76(2), pages 285-297, August.
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    Cited by:

    1. José García & Román Salmerón & Catalina García & María del Mar López Martín, 2016. "Standardization of Variables and Collinearity Diagnostic in Ridge Regression," International Statistical Review, International Statistical Institute, vol. 84(2), pages 245-266, August.

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