Warped Wavelet and Vertical Thresholding

TL;DR

This paper introduces a new adaptive wavelet-based estimator for nonparametric regression with random design, achieving optimal convergence rates by leveraging warped wavelets and hierarchical thresholding.

Contribution

It proposes a novel estimator using warped wavelets and a tree-like thresholding scheme that adapts to the design distribution and achieves optimal convergence rates.

Findings

Estimator is adaptive to the design distribution.

Achieves optimal convergence rates in $L^2$ norm.

Provides a computationally feasible implementation.

Abstract

Let $\{(X_i,Y_i)\}_{i\in \{1,..., n\}}$ be an i.i.d. sample from the random design regression model $Y=f(X)+\epsilon$ with $(X,Y)\in [0,1]\times [-M,M]$. In dealing with such a model, adaptation is naturally to be intended in terms of $L^2([0,1],G_X)$ norm where $G_X(\cdot)$ denotes the (known) marginal distribution of the design variable $X$. Recently much work has been devoted to the construction of estimators that adapts in this setting (see, for example, [5,24,25,32]), but only a few of them come along with a easy--to--implement computational scheme. Here we propose a family of estimators based on the warped wavelet basis recently introduced by Picard and Kerkyacharian [36] and a tree-like thresholding rule that takes into account the hierarchical (across-scale) structure of the wavelet coefficients. We show that, if the regression function belongs to a certain class of approximation spaces defined in terms of $G_X(\cdot)$, then our procedure is adaptive and converge to the true regression function with an optimal rate. The results are stated in terms of excess probabilities as in [19].