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July 2004 Weighted uniform consistency of kernel density estimators
Evarist Giné, Vladimir Koltchinskii, Joel Zinn
Ann. Probab. 32(3B): 2570-2605 (July 2004). DOI: 10.1214/009117904000000063

Abstract

Let fn denote a kernel density estimator of a continuous density f in d dimensions, bounded and positive. Let Ψ(t) be a positive continuous function such that ‖Ψfβ<∞ for some 0<β<1/2. Under natural smoothness conditions, necessary and sufficient conditions for the sequence ${\sqrt{\frac{nh_{n}^{d}}{2|\log h_{n}^{d}|}}\|\Psi(t)(f_{n}(t)-Ef_{n}(t))\|_{\infty}}$ to be stochastically bounded and to converge a.s. to a constant are obtained. Also, the case of larger values of β is studied where a similar sequence with a different norming converges a.s. either to 0 or to +∞, depending on convergence or divergence of a certain integral involving the tail probabilities of Ψ(X). The results apply as well to some discontinuous not strictly positive densities.

Citation

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Evarist Giné. Vladimir Koltchinskii. Joel Zinn. "Weighted uniform consistency of kernel density estimators." Ann. Probab. 32 (3B) 2570 - 2605, July 2004. https://doi.org/10.1214/009117904000000063

Information

Published: July 2004
First available in Project Euclid: 6 August 2004

zbMATH: 1052.62034
MathSciNet: MR2078551
Digital Object Identifier: 10.1214/009117904000000063

Subjects:
Primary: 62G07
Secondary: 60F15 , 62G20

Keywords: kernel density estimator , rates of convergence , weak and strong weighted uniform consistency , weighted L∞-norm

Rights: Copyright © 2004 Institute of Mathematical Statistics

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Vol.32 • No. 3B • July 2004
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