Robust Estimation and Inference for Semiparametric and Nonparametric Regression Models

Abstract

Parametric regression methods are efficient when correctly specified but are sensitive to model misspecification and outliers. Nonparametric regression offers greater flexibility at the cost of reduced interpretability and susceptibility to the curse of dimensionality. Semiparametric models provide a compromise between these approaches by combining structural interpretability with functional flexibility. A key limitation of many classical semiparametric and nonparametric methods, however, is their lack of robustness to heavy-tailed errors and contaminated data. In this paper, we develop robust kernel, spline, and single-index regression estimators based on robust loss functions. To facilitate inference, we propose bootstrap-based procedures that remain valid in settings where classical assumptions may be violated. Through extensive simulation studies under normal, heavy-tailed, and contaminated error distributions, we demonstrate that the proposed robust methods achieve comparable performance to classical approaches in clean settings while providing substantial gains in stability and inferential reliability under contamination. Unlike existing works that study these robust estimators in isolation, the proposed approach provides a unified framework that integrates robust kernel regression, robust spline regression, and robust single-index modeling with a coherent bootstrap-based inference procedure. Application to Boston housing data further illustrates the practical usefulness of the proposed methodology.