A Bias-Corrected Cp Criterion for Optimizing Ridge Parameters in Multivariate Generalized Ridge Regression

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  • Yanagihara Hirokazu
    Department of Mathematics, Graduate School of Science, Hiroshima University
  • Nagai Isamu
    Department of Mathematics, Graduate School of Science, Hiroshima University
  • Satoh Kenichi
    Department of Environmetrics and Biometrics, Research Institute for Radiation Biology and Medicine, Hiroshima University

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Other Title
  • 多変量一般化リッジ回帰におけるリッジパラメータ最適化のためのバイアス補正Ｃｐ規準
  • タヘンリョウ イッパンカ リッジ カイキ ニ オケル リッジパラメータ サイテキカ ノ タメ ノ バイアス ホセイ Cp キジュン

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Abstract

In a ridge regression for an univariate linear regression model, it is common that an optimal ridge parameter is determined by minimizing an information criterion, e.g., Mallows' Cp criterion (Mallows (1973, 1995)). Since the solution to the minimization problem of the information criterion is not expressed by a closed form, an additional computational task is required. On the other hand, a generalized ridge regression proposed by Hoerl and Kennard (1970) has multiple ridge parameters, but optimal ridge parameters are obtained by closed forms. In this paper, we extend the generalized ridge regression to a multivariate linear regression case. Then, Cp criterion for optimizing ridge parameters in the multivariate generalized ridge regression is considered as an estimator of a risk function based on the mean square error of prediction. By correcting a bias of the Cp criterion completely, a bias-corrected Cp criterion named by modified Cp (MCp) criterion is proposed. It is analytically proved that the proposed MCp has not only smaller bias but also smaller variance than an existing Cp criterion and is the uniformly minimum variance unbiased estimator of the risk function. We show that the criterion has useful properties by means of numerical experiments.

Journal

  • Ouyou toukeigaku

    Ouyou toukeigaku 38 (3), 151-172, 2009

    Japanese Society of Applied Statistics

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