UNIFORM LIMIT THEOREMS FOR WAVELET DENSITY ESTIMATORS

• 成果类型: Article
• 署名作者: Gine, Evarist; Nickl, Richard
• 署名单位: University of Connecticut; University of Cambridge
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/08-AOP447
• 发表日期: 2009
• 页码: 1605-1646
• 关键词: Kernel; Consistency; rates; inequalities

摘要

Let p(n)(y) = Sigma(k)(alpha) over cap (k)phi(y - k) + Sigma(jn-1)(l=0) Sigma(k)(beta) over cap (lk)2(l/2)psi(2(l) y-k) be the linear wavelet density estimator, where phi, psi are a father and a mother wavelet (with compact support), (alpha) over cap (k), (beta) over cap (lk) are the empirical wavelet coefficients based on an i.i.d. sample of random variables distributed according to a density p(0) on R, and j(n) is an element of Z, j(n) NE arrow infinity. Several uniform limit theorems are proved: First, the almost sure rate of convergence of sup(y is an element of R) | p(n) (y) - Ep(n) (y) | is obtained, and a law of the logarithm for a suitably scaled version of this quantity is established. This implies that sup(y is an element of R) |p(n)(y) - p(0)(y)| attains the optimal almost sure rate of convergence for estimating p(0), if j(n) is suitably chosen. Second, a uniform central limit theorem as well as strong invariance principles for the distribution function of p(n), that is, for the stochastic processes root n(F-n(W)(s) - F(s)) = root n integral(s)(-infinity) (p(n) - p(0)), s is an element of R, are proved; and more generally, uniform central limit theorems for the processes root n integral(p(n) - p(0))f, f is an element of F, for other Donsker classes F of interest are considered. As a statistical application, it is shown that essentially the same limit theorems can be obtained for the hard thresholding wavelet estimator introduced by Donoho et al. [Ann. Statist. 24 (1996) 508-539].