Uniform limit theorems for wavelet density estimators

Abstract

Let $p_n(y)={\sum }_k{\hat{\alpha }}_k\varphi (y-k)+{\sum }_{l=0}^{j_n-1}{\sum }_k{\hat{\beta }}_{lk}{2}^{l/2}\psi (2^{l}y-k)$ be the linear wavelet density estimator, where $\varphi$, $\psi$ are a father and a mother wavelet (with compact support), ${\hat{\alpha }}_k$, ${\hat{\beta }}_{lk}$ are the empirical wavelet coefficients based on an i.i.d. sample of random variables distributed according to a density $p_0$ on $\mathbb{R}$, and $j_n\in \mathbb{Z}$, $j_n\nearrow \infty$. Several uniform limit theorems are proved: First, the almost sure rate of convergence of ${\sup }_{y\in \mathbb{R}}|p_n(y)-E{p}_n(y)|$ is obtained, and a law of the logarithm for a suitably scaled version of this quantity is established. This implies that ${\sup }_{y\in \mathbb{R}}|p_n(y)-p_0(y)|$ 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 $\sqrt{n}(F_n^W(s)-F(s))=\sqrt{n}{\int }_{-\infty }^s(p_n-p_0),s\in \mathbb{R}$, are proved; and more generally, uniform central limit theorems for the processes $\sqrt{n}\int (p_n-p_0)f$, $f\in \mathcal{F}$, for other Donsker classes $\mathcal{F}$ 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].

Published in at http://dx.doi.org/10.1214/08-AOP447 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)

References in corpus (3)

• Uniform in bandwidth consistency of kernel-type function estimators
• Exponential and moment inequalities for U-statistics
• Empirical and Gaussian processes on Besov classes

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