A robust spline approach in partially linear additive models

TL;DR

This paper introduces a robust spline-based estimation method for partially linear additive models that effectively handles outliers, providing reliable inference and improved performance over classical methods in contaminated data scenarios.

Contribution

It develops a new family of robust estimators combining B-splines with robust regression, with theoretical guarantees and practical validation.

Findings

Robust estimators outperform classical methods under data contamination.

The proposed method achieves consistency and asymptotic normality.

Numerical experiments confirm improved finite-sample performance.

Abstract

Partially linear additive models generalize linear ones since they model the relation between a response variable and covariates by assuming that some covariates have a linear relation with the response but each of the others enter through unknown univariate smooth functions. The harmful effect of outliers either in the residuals or in the covariates involved in the linear component has been described in the situation of partially linear models, that is, when only one nonparametric component is involved in the model. When dealing with additive components, the problem of providing reliable estimators when atypical data arise, is of practical importance motivating the need of robust procedures. Hence, we propose a family of robust estimators for partially linear additive models by combining $B-$splines with robust linear regression estimators. We obtain consistency results, rates of convergence and asymptotic normality for the linear components, under mild assumptions. A Monte Carlo study is carried out to compare the performance of the robust proposal with its classical counterpart under different models and contamination schemes. The numerical experiments show the advantage of the proposed methodology for finite samples. We also illustrate the usefulness of the proposed approach on a real data set.