Some asymptotic results on density estimators by wavelet projections

TL;DR

This paper investigates the asymptotic behavior of wavelet-based density estimators, establishing exact convergence rates and conditions under which they fail to converge uniformly.

Contribution

It provides precise almost sure convergence rates for wavelet density estimators and identifies conditions affecting their uniform consistency.

Findings

Exact rates of almost sure convergence when $n2^{-dj_n}/ ext{log} n o ext{infinity}$.

Failure of uniform convergence when $n2^{-dj_n}/ ext{log} n$ converges to a positive constant.

Conditions under which the wavelet density estimators are consistent or inconsistent.

Abstract

Let $(X_i)_{i\geq 1}$ be an i.i.d. sample on $\RRR^d$ having density $f$. Given a real function $\phi$ on $\RRR^d$ with finite variation and given an integer valued sequence $(j_n)$, let $\fn$ denote the estimator of $f$ by wavelet projection based on $\phi$ and with multiresolution level equal to $j_n$. We provide exact rates of almost sure convergence to 0 of the quantity $\sup_{x\in H}\mid \fn(x)-\EEE(\fn)(x)\mid$, when $n2^{-dj_n}/\log n \rar \infty$ and $H$ is a given hypercube of $\RRR^d$. We then show that, if $n2^{-dj_n}/\log n \rar c$ for a constant $c>0$, then the quantity $\sup_{x\in H}\mid \fn(x)-f\mid$ almost surely fails to converge to 0.