Wavelet-based robust estimation and variable selection in nonparametric additive models

Umberto Amato,Irène Gijbels,Italia De Feis,Anestis Antoniadis

doi:10.1007/s11222-021-10065-z

Abstract

This article studies M-type estimators for fitting robust additive models in the presence of anomalous data. The components in the additive model are allowed to have different degrees of smoothness. We introduce a new class of wavelet-based robust M-type estimators for performing simultaneous additive component estimation and variable selection in such inhomogeneous additive models. Each additive component is approximated by a truncated series expansion of wavelet bases, making it feasible to apply the method to nonequispaced data and sample sizes that are not necessarily a power of 2. Sparsity of the additive components together with sparsity of the wavelet coefficients within each component (group), results into a bi-level group variable selection problem. In this framework, we discuss robust estimation and variable selection. A two-stage computational algorithm, consisting of a fast accelerated proximal gradient algorithm of coordinate descend type, and thresholding, is proposed. When using nonconvex redescending loss functions, and appropriate nonconvex penalty functions at the group level, we establish optimal convergence rates of the estimates. We prove variable selection consistency under a weak compatibility condition for sparse additive models. The theoretical results are complemented with some simulations and real data analysis, as well as a comparison to other existing methods.

Highlights

Additive regression models have turned out to be useful statistical tools in the analysis of high-dimensional data
We propose a new class of waveletbased robust M-type estimators for performing simultaneous additive component estimation and variable selection in such sparse “inhomogeneous” additive models
Component 2 −3 −2 −1 0 1 2 3 complemented the simulations with two similar scenarios but with an equidistant deterministic design of size a power of 2 allowing us to consider the robust RAMlet nonlinear back-fitting wavelet-based estimator of additive models developed in Sardy and Tseng (2004), in addition to the alternatives used in scenarios 1 and 2

Summary

Introduction

Additive regression models have turned out to be useful statistical tools in the analysis of high-dimensional data. The attraction of such models is that the additive components can be estimated with the same optimal convergence rate as a one-dimensional nonparametric regression. Some extreme observations may occur, and estimation using an unbounded loss function suffers from a lack of robustness, meaning that the estimated functions can be distorted by the outliers. Both the nonparametric function estimates themselves and the choice of the penalization parameters associated with them are affected

Methods

Results

Conclusion