On the integrated squared error of the linear wavelet density estimator

TL;DR

This paper analyzes the asymptotic behavior of linear wavelet density estimators, establishing laws of the iterated logarithm and Berry-Esseen type theorems under less restrictive smoothness conditions than kernel density estimators.

Contribution

It introduces new theoretical results on the asymptotic properties of wavelet density estimators, with relaxed smoothness assumptions compared to traditional kernel methods.

Findings

Established law of the iterated logarithm for wavelet density estimators.

Proved a Berry-Esseen type theorem with convergence rates.

Identified conditions on the density for asymptotic variance calculation.

Abstract

Linear wavelet density estimators are wavelet projections of the empirical measure based on independent, identically distributed observations. We study here the law of the iterated logarithm (LIL) and a Berry-Esseen type theorem. These results are proved under different assumptions on the density $f$ that are different from those needed for similar results in the case of convolution kernels (KDE): whereas the smoothness requirements are much less stringent than for the KDE, Riemann integrability assumptions are needed in order to compute the asymptotic variance, which gives the scaling constant in LIL. To study the Berry-Esseen type theorem, a rate of convergence result in the martingale CLT is used.