The Annals of Statistics

Adaptive Multivariate Ridge Regression

P. J. Brown and J. V. Zidek

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A multivariate version of the Hoerl-Kennard ridge regression rule is introduced. The choice from among a large class of possible generalizations is guided by Bayesian considerations; the result is implicitly in the work of Lindley and Smith although not actually derived there. The proposed rule, in a variety of equivalent forms is discussed and the choice of its ridge matrix considered. As well, adaptive multivariate ridge rules and closely related empirical Bayes procedures are presented, these being for the most part formal extensions of certain univariate rules. Included is the Efron-Morris multivariate version of the James-Stein estimator. By means of an appropriate generalization of a result of Morris (see Thisted) the mean square error of these adaptive and empirical Bayes rules are compared.

Article information

Ann. Statist., Volume 8, Number 1 (1980), 64-74.

First available in Project Euclid: 12 April 2007

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Zentralblatt MATH identifier


Primary: 62J05: Linear regression
Secondary: 62C99: None of the above, but in this section 62H99: None of the above, but in this section 62C10: Bayesian problems; characterization of Bayes procedures 62C15: Admissibility

Empirical Bayes minimax unbiased risk estimation Bayesian regression


Brown, P. J.; Zidek, J. V. Adaptive Multivariate Ridge Regression. Ann. Statist. 8 (1980), no. 1, 64--74. doi:10.1214/aos/1176344891.

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