Uniform central limit theorems for Kernel density estimators

• 成果类型: Article
• 署名作者: Gine, Evarist; Nickl, Richard
• 署名单位: University of Connecticut
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-007-0087-9
• 发表日期: 2008
• 页码: 333-387
• 关键词: WEAK-CONVERGENCE; donsker classes

摘要

Let Pn* K-hn (x) = n(-1)h(n)(-d) Sigma(n)(i= 1) K (( x - X-i)/h(n)) be the classical kernel density estimator based on a kernel K and n independent random vectors X-i each distributed according to an absolutely continuous law P on R-d. It is shown that the processes f ->root n f d(P-n * K-hn - P), f is an element of F, converge in law in the Banach space l infinity(F), for many interesting classes F of functions or sets, some P- Donsker, some just P- pregaussian. The conditions allow for the classical bandwidths hn that simultaneously ensure optimal rates of convergence of the kernel density estimator in mean integrated squared error, thus showing that, subject to some natural conditions, kernel density estimators are 'plug- in' estimators in the sense of Bickel and Ritov (Ann Statist 31: 1033 - 1053, 2003). Some new results on the uniform central limit theorem for smoothed empirical processes, needed in the proofs, are also included.