Least Squares Wavelet-based Estimation for Additive Regression Models using Non Equally-Spaced Designs

TL;DR

This paper introduces a wavelet-based least squares method for additive regression models that works with non-equally spaced data, achieving strong consistency and optimal convergence rates without requiring equispaced designs.

Contribution

It develops a novel wavelet-based estimator for additive models that is consistent and rate-optimal even with non-uniform data sampling, expanding applicability.

Findings

Estimator achieves strong consistency in L2 norm

Convergence rates are optimal up to a logarithmic factor

Practical effectiveness demonstrated through simulations and real data

Abstract

Additive regression models are actively researched in the statistical field because of their usefulness in the analysis of responses determined by non-linear relationships with multivariate predictors. In this kind of statistical models, the response depends linearly on unknown functions of predictor variables and typically, the goal of the analysis is to make inference about these functions. In this paper, we study the problem of additive regression using a least squares approach based on periodic orthogonal wavelets on the interval [0,1]. For this estimator, we obtain strong consistency (with respect to the $\mathbb{L}_{2}$ norm) characterized by optimal convergence rates up to a logarithmic factor, independent of the dimensionality of the problem. This is achieved by truncating the model estimates by a properly chosen parameter, and selecting the multiresolution level $J$ used for the wavelet expansion, as a function of the sample size. In this approach, we obtain these results without the assumption of an equispaced design, a condition that is typically assumed in most wavelet-based procedures. Finally, we show practical results obtained from a simulation study and a real life application, demonstrating the applicability of the proposed methods for the problem of non-linear robust additive regression models.