Open Access
December 1997 Nonasymptotic universal smoothing factors, kernel complexity and Yatracos classes
Luc Devroye, Gábor Lugosi
Ann. Statist. 25(6): 2626-2637 (December 1997). DOI: 10.1214/aos/1030741088

Abstract

We introduce a method to select a smoothing factor for kernel density estimation such that, for all densities in all dimensions, the $L_1$ error of the corresponding kernel estimate is not larger than three times the error of the estimate with the optimal smoothing factor plus a constant times $\sqrt{\log n/n}$, where n is the sample size, and the constant depends only on the complexity of the kernel used in the estimate. The result is nonasymptotic, that is, the bound is valid for each n. The estimate uses ideas from the minimum distance estimation work of Yatracos. As the inequality is uniform with respect to all densities, the estimate is asymptotically minimax optimal (modulo a constant) over many function classes.

Citation

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Luc Devroye. Gábor Lugosi. "Nonasymptotic universal smoothing factors, kernel complexity and Yatracos classes." Ann. Statist. 25 (6) 2626 - 2637, December 1997. https://doi.org/10.1214/aos/1030741088

Information

Published: December 1997
First available in Project Euclid: 30 August 2002

zbMATH: 0897.62035
MathSciNet: MR1604428
Digital Object Identifier: 10.1214/aos/1030741088

Subjects:
Primary: 62G05

Keywords: asymptotic optimality , convergence , Density estimation , kernel estimate , minimum distance estimate , smoothing factor

Rights: Copyright © 1997 Institute of Mathematical Statistics

Vol.25 • No. 6 • December 1997
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