Weighted uniform consistency of kernel density estimators

• 成果类型: Article
• 署名作者: Giné, E; Koltchinskii, V; Zinn, J
• 署名单位: University of Connecticut; University of Connecticut; University of New Mexico; Texas A&M University System; Texas A&M University College Station; University of Washington; University of Washington Seattle; Sorbonne Universite
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/009117904000000063
• 发表日期: 2004
• 页码: 2570-2605
• 关键词: limit-theorems; rates; logarithm; bounds; LAW

摘要

Let fin denote a kernel density estimator of a continuous density f in d dimensions, bounded and positive. Let Psi(t) be a positive continuous function such that parallel toPsif(beta)parallel toinfinity < infinity for some 0 < beta < 1/2. Under natural smoothness conditions, necessary and sufficient conditions for the sequence rootnh(n)(d)/2\logh(n)(d)\ parallel toPsi(t)(f(n)(t) - Efn(t))parallel toinfinity to be stochastically bounded and to converge a.s. to a constant are obtained. Also, the case of larger values of beta is studied where a similar sequence with a different norming converges a.s. either to 0 or to +infinity, depending on convergence or divergence of a certain integral involving the tail probabilities of Psi(X). The results apply as well to some discontinuous not strictly positive densities.