Weighted uniform consistency of kernel density estimators
Evarist Giné, Vladimir Koltchinskii, and Joel Zinn
Source: Ann. Probab. Volume 32, Number 3B
(2004), 2570-2605.
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.
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Keywords: Kernel density estimator; rates of convergence; weak and strong weighted uniform consistency; weighted L∞-norm
Full-text: Open access
Links and Identifiers
Permanent link to this document: http://projecteuclid.org/euclid.aop/1091813624
Digital Object Identifier: doi:10.1214/009117904000000063
Zentralblatt MATH identifier: 02121707
Mathematical Reviews number (MathSciNet): MR2078551
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