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An unbiased Cp criterion for multivariate ridge regression

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  • Yanagihara, Hirokazu
  • Satoh, Kenichi

Abstract

Mallows' Cp statistic is widely used for selecting multivariate linear regression models. It can be considered to be an estimator of a risk function based on an expected standardized mean square error of prediction. An unbiased Cp criterion for selecting multivariate linear regression models has been proposed. In this paper, that unbiased Cp criterion is extended to the case of a multivariate ridge regression. It is analytically proved that the proposed criterion has not only a smaller bias but also a smaller variance than the 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.

Suggested Citation

  • Yanagihara, Hirokazu & Satoh, Kenichi, 2010. "An unbiased Cp criterion for multivariate ridge regression," Journal of Multivariate Analysis, Elsevier, vol. 101(5), pages 1226-1238, May.
    • Handle: RePEc:eee:jmvana:v:101:y:2010:i:5:p:1226-1238
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    References listed on IDEAS

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    1. Bradley Efron, 2004. "The Estimation of Prediction Error: Covariance Penalties and Cross-Validation," Journal of the American Statistical Association, American Statistical Association, vol. 99, pages 619-632, January.
    2. Yasunori Fujikoshi & Takafumi Noguchi & Megu Ohtaki & Hirokazu Yanagihara, 2003. "Corrected versions of cross-validation criteria for selecting multivariate regression and growth curve models," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 55(3), pages 537-553, September.
    3. Clifford M. Hurvich & Jeffrey S. Simonoff & Chih‐Ling Tsai, 1998. "Smoothing parameter selection in nonparametric regression using an improved Akaike information criterion," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 60(2), pages 271-293.
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    Cited by:

    1. Mori, Yuichi & Suzuki, Taiji, 2018. "Generalized ridge estimator and model selection criteria in multivariate linear regression," Journal of Multivariate Analysis, Elsevier, vol. 165(C), pages 243-261.
    2. Ogasawara, Haruhiko, 2015. "Distribution-free properties of some asymptotic cumulants for the Mallows Cp and its modifications," ビジネス創造センターディスカッション・ペーパー (Discussion papers of the Center for Business Creation) 10252/5499, Otaru University of Commerce.
    3. Fujikoshi, Yasunori & Sakurai, Tetsuro & Yanagihara, Hirokazu, 2014. "Consistency of high-dimensional AIC-type and Cp-type criteria in multivariate linear regression," Journal of Multivariate Analysis, Elsevier, vol. 123(C), pages 184-200.

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