The Annals of Probability

Weighted uniform consistency of kernel density estimators

Evarist Giné, Vladimir Koltchinskii, and Joel Zinn

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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

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

First available in Project Euclid: 6 August 2004

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

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


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.

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