## The Annals of Probability

### Weighted uniform consistency of kernel density estimators

#### 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 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

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

#### 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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