Spatially Adaptive Density Estimation by Localised Haar Projections

TL;DR

This paper introduces Haar wavelet estimators with localised test procedures that adapt to varying smoothness of an unknown density function across different spatial regions, achieving optimal sup-norm convergence.

Contribution

It presents a novel adaptive density estimation method using localised Haar projections that adjust resolution levels based on local smoothness, with practical thresholding rules.

Findings

Estimators adapt to spatially heterogeneous smoothness.

Achieve optimal sup-norm convergence rates.

Practical thresholding rules justified by information theory.

Abstract

Given a random sample from some unknown density $f_0: \mathbb R \to [0, \infty)$ we devise Haar wavelet estimators for $f_0$ with variable resolution levels constructed from localised test procedures (as in Lepski, Mammen, and Spokoiny (1997, Ann. Statist.)). We show that these estimators adapt to spatially heterogeneous smoothness of $f_0$, simultaneously for every point $x$ in a fixed interval, in sup-norm loss. The thresholding constants involved in the test procedures can be chosen in practice under the idealised assumption that the true density is locally constant in a neighborhood of the point $x$ of estimation, and an information theoretic justification of this practice is given.