Weighted uniform consistency of kernel density estimators

Giné, Evarist ; Koltchinskii, Vladimir ; Zinn, Joel

Ann. Probab., Tome 32 (2004) no. 1A, p. 2570-2605 / Harvested from Project Euclid

Résumé

Let f_n 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 ‖Ψf^β‖_∞<∞ 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.

Détail

Publié le : 2004-07-14
Classification:  Kernel density estimator,  rates of convergence,  weak and strong weighted uniform consistency,  weighted L∞-norm,  62G07,  60F15,  62G20