Uniform Limit Theorems for Wavelet Density Estimators

Let $p_{n}(y) = \Sigma_{k}\hat{\alpha}_{k}\phi(y - k) + \Sigma_{l=0}^{j_{n}-1} \Sigma_{k} \hat{\beta}_{lk}2^{l/2}\psi(2^{l}y - k)$ be the linear wavelet density estimator, where ϕ, ψ 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 $j_{n} \epsilon \mathbb{Z}$ , $j_{n} \nearrow \infty$ . Several uniform limit theorems are proved: First, the almost sure rate of convergence of $sup_{y\epsilon\mathbb{R}}|p_{n}(y) - Ep_{n}(y)|$ is obtained, and a law of the logarithm for a suitably scaled version of this quantity is established. This implies that $sup_{y\epsilon\mathbb{R}}|p_{n}(y) - p_{0}(y)|$ 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 $\sqrt{n}(F_{n}^{W}(s) - F(s)) = \sqrt{n} \int_{-\infty}^{s}(p_{n} - p_{0}), s \epsilon \mathbb{R}$ , are proved; and more generally, uniform central limit theorems for the processes $\sqrt{n}\int(p_{n} - p_{0})f$ , $f \epsilon \digamma$ , for other Donsker classes $\digamma$ 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].