The Annals of Statistics

Consistent Cross-Validated Density Estimation

Y.-S. Chow, S. Geman, and L.-D. Wu

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Application of nonparametric density estimators generally requires the specification of a "smoothing parameter." The kernel estimator, for example, is not fully defined until a window width, or scaling, for the kernels has been chosen. Many "data-driven" techniques have been suggested for the practical choice of smoothing parameter. Of these, the most widely studied is the method of cross-validation. Our own simulations, as well as those of many other investigators, indicate that cross-validated smoothing can be an extremely effective practical solution. However, many of the most basic properties of cross-validated estimators are unknown. Indeed, recent results show that cross-validated estimators can fail even to be consistent for seemingly well-behaved problems. In this paper we will review the application of cross-validation to the smoothing problem, and establish $L_1$ consistency for certain cross-validated kernels and histograms.

Article information

Ann. Statist., Volume 11, Number 1 (1983), 25-38.

First available in Project Euclid: 12 April 2007

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


Primary: 62G05: Estimation
Secondary: 62A10

Cross-validation consistency nonparametric density estimation


Chow, Y.-S.; Geman, S.; Wu, L.-D. Consistent Cross-Validated Density Estimation. Ann. Statist. 11 (1983), no. 1, 25--38. doi:10.1214/aos/1176346053.

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