Development and Assessment of a Robust Nonparametric Regression Method

Thesis Abstract

In contemporary statistical analysis, nonparametric regression techniques have gained prominence due to their flexibility in modeling complex, nonlinear relationships without strict distributional assumptions. Despite their advantages, existing nonparametric methods often exhibit sensitivity to outliers and heteroscedasticity, compromising the robustness and reliability of the estimates in real-world datasets. This study aims to develop and empirically evaluate a novel robust nonparametric regression method designed to enhance resilience against anomalies while maintaining estimation efficiency. The specific objectives include (1) formulating a robust nonparametric regression approach utilizing a combination of kernel smoothing and robust loss functions; (2) implementing the proposed method within a computational framework; (3) comparing its performance to existing techniques such as local polynomial regression and smoothing splines in both simulated and real datasets; and (4) assessing its robustness, accuracy, and computational efficiency through comprehensive empirical analysis. The research adopts a mixed-methods approach. Quantitative data is collected through simulations, where datasets are generated with known underlying functions and controlled levels of contamination to evaluate the method's robustness effectively. Additionally, real-world data comprising 500 observations from environmental monitoring datasets—characterized by potential outliers and variability—is employed to validate practical applicability. The development phase involves deriving the mathematical formulation of the proposed estimator, based on robust loss functions such as Huber’s loss, coupled with adaptive kernel bandwidth selection, guided by theoretical underpinnings from the framework of M-estimators. Implementation is carried out via programming in R, utilizing packages like ‘np’ for nonparametric methods, with custom modifications to integrate robustness features. Data analysis involves the use of simulation studies to compare bias, mean squared error (MSE), and robustness metrics across methods. Metrics such as breakdown point and influence function are computed to quantitatively assess robustness. For empirical data, techniques include residual analysis, diagnostic plots, and cross-validation to evaluate model fit and prediction accuracy. The theoretical justification of the method draws upon the foundations of robust statistics, particularly the theories of Huber and Tukey, with the empirical performance evaluated through statistical significance tests and comparative analysis. Anticipated findings suggest that the proposed robust nonparametric regression method will outperform traditional approaches in contaminated datasets by demonstrating lower bias and MSE, higher efficiency in the presence of outliers, and comparable computational costs. It is expected that the new approach will exhibit superior breakdown points and influence functions, indicating enhanced robustness. The study aims to establish the method as a viable alternative for practitioners requiring reliable nonlinear modeling in the presence of data abnormalities. This research contributes to the advancement of nonparametric regression methodology by introducing a robust estimator that blends flexibility with resilience, addressing gaps identified in existing literature regarding outlier sensitivity. The findings are expected to inform both theoretical developments in robust statistics and practical applications across fields such as environmental science, econometrics, and biomedical research. The main conclusion is that integrating robust loss functions with kernel-based smoothing techniques significantly improves the performance of nonparametric regression in contaminated data scenarios. Recommendations include the adoption of the proposed method in scenarios characterized by outliers and heteroscedasticity, further refinement through adaptive bandwidth strategies, and extension to multivariate settings. Future research should explore scalable algorithms for high-dimensional data and investigate the integration of machine learning techniques to automatically tune robustness parameters, thereby broadening the applicability of the proposed approach in complex, large-scale datasets.