The Annals of Probability

Weighted uniform consistency of kernel density estimators

Evarist Giné, Vladimir Koltchinskii, and Joel Zinn

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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.

Article information

Source
Ann. Probab., Volume 32, Number 3B (2004), 2570-2605.

Dates
First available in Project Euclid: 6 August 2004

Permanent link to this document
https://projecteuclid.org/euclid.aop/1091813624

Digital Object Identifier
doi:10.1214/009117904000000063

Mathematical Reviews number (MathSciNet)
MR2078551

Zentralblatt MATH identifier
1052.62034

Subjects
Primary: 62G07: Density estimation
Secondary: 60F15: Strong theorems 62G20: Asymptotic properties

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

Citation

Giné, Evarist; Koltchinskii, Vladimir; Zinn, Joel. Weighted uniform consistency of kernel density estimators. Ann. Probab. 32 (2004), no. 3B, 2570--2605. doi:10.1214/009117904000000063. https://projecteuclid.org/euclid.aop/1091813624


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