Hiroshima Mathematical Journal

Optimization of ridge parameters in multivariate generalized ridge regression by plug-in methods

Isamu Nagai, Hirokazu Yanagihara, and Kenichi Satoh

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Generalized ridge (GR) regression for an univariate linear model was proposed simultaneously with ridge regression by Hoerl and Kennard (1970). In this paper, we deal with a GR regression for a multivariate linear model, referred to as a multivariate GR (MGR) regression. From the viewpoint of reducing the mean squared error (MSE) of a predicted value, many authors have proposed several GR estimators consisting of ridge parameters optimized by non-iterative methods. By expanding their optimizations of ridge parameters to the multiple response case, we derive some MGR estimators with ridge parameters optimized by the plug-in method. We analytically compare obtained MGR estimators with existing MGR estimators, and numerical studies are also given for an illustration.

Article information

Hiroshima Math. J., Volume 42, Number 3 (2012), 301-324.

First available in Project Euclid: 11 December 2012

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62J07: Ridge regression; shrinkage estimators
Secondary: 62H12: Estimation

Generalized ridge regression Mallows’ $C_p$ statistic model selection multivariate linear regression model non-iterative estimation plug-in method


Nagai, Isamu; Yanagihara, Hirokazu; Satoh, Kenichi. Optimization of ridge parameters in multivariate generalized ridge regression by plug-in methods. Hiroshima Math. J. 42 (2012), no. 3, 301--324. doi:10.32917/hmj/1355238371. https://projecteuclid.org/euclid.hmj/1355238371

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