Weighted uniform consistency of kernel density estimators
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Evarist Giné [1] ; Vladimir Koltchinskii [2] ; Joel Zinn [3]
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[1] University of Connecticut
University of Connecticut
Town of Mansfield, Estados Unidos
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[2] University of New Mexico
University of New Mexico
Estados Unidos
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[3] A&M University
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- Localización: Annals of probability: An official journal of the Institute of Mathematical Statistics, ISSN 0091-1798, Vol. 32, Nº. 3, 2, 2004, págs. 2570-2605
- Idioma: inglés
- DOI: 10.1214/009117904000000063
- Texto completo no disponible (Saber más ...)
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Resumen
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.
