Open Access
Translator Disclaimer
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


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.


Download Citation

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


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

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

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


Vol.32 • No. 3B • July 2004
Back to Top