Adaptive estimation of a distribution function and its density in sup-norm loss by wavelet and spline projections

TL;DR

This paper introduces adaptive, data-driven estimators for distribution functions and densities that achieve optimal convergence rates in sup-norm loss using wavelet and spline projections, with a novel thresholding method.

Contribution

It develops a new adaptive estimation procedure combining model selection with Rademacher process-based thresholds for improved sup-norm convergence rates.

Findings

Estimators achieve optimal sup-norm rates for distribution and density estimation.

Method effectively adapts to unknown smoothness of the underlying density.

The approach extends to wavelet and spline projection frameworks.

Abstract

Given an i.i.d. sample from a distribution $F$ on $\mathbb{R}$ with uniformly continuous density $p_0$, purely data-driven estimators are constructed that efficiently estimate $F$ in sup-norm loss and simultaneously estimate $p_0$ at the best possible rate of convergence over H\"older balls, also in sup-norm loss. The estimators are obtained by applying a model selection procedure close to Lepski's method with random thresholds to projections of the empirical measure onto spaces spanned by wavelets or $B$-splines. The random thresholds are based on suprema of Rademacher processes indexed by wavelet or spline projection kernels. This requires Bernstein-type analogs of the inequalities in Koltchinskii [Ann. Statist. 34 (2006) 2593-2656] for the deviation of suprema of empirical processes from their Rademacher symmetrizations.