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Biometrika 1995 82(2):418-425; doi:10.1093/biomet/82.2.418
© 1995 by Biometrika Trust
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MISCELLANEA

Relative rates of convergence for efficient model selection criteria in linear regression

CLIFFORD M. HURVICH and CHIN-LING TSAI

Department of Statistics and Operations Research, New York University 44 West Fourth Street, New York, New York 10012, U.S.A.
Graduate School of Management, University of California Davis, California 95616, U.S.A.

Two approaches to estimating a smooth regression function not specified by a finite number of parameters are (i) to fit a parametric model to the data, with the number of parameters selected by the Akaike information criterion, AIC, and (ii) to use nonparametric smoothing techniques, with the smoothing parameter selected by cross-validation or some other automatic method. We consider the relative rate of convergence of the mean integrated squared error of the selected estimator compared to the best possible estimator in the class of candidates under consideration. We extend results of Shibata (1981) to show that the relative rate of convergence using AIC in the parametric case can be as fast as op(n–1/2+{xi}) for arbitrary positive {xi}. We also show that this rate can be attained in the special cases of polynomial and trigonometric regression.

Key Words: Akaike information criterion • Bandwidth selection • Nonparametric regression


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