Robust Statistical Modeling In Functional Linear Regression

Functional linear regression is a prominent field within the domain of functional data analysis, with extensive applications in various domains such as biomedical studies, brain imaging, and chemometrics. However, despite the abundance of literature on functional linear regression, limited attention has been devoted to addressing outliers or heavy-tailed distributions in the data. Consequently, robust statistical analysis remains an underdeveloped practice in this area. The primary objective of this dissertation is to enhance the utilization of robust methods for modeling functional linear regression by primarily focusing on robust estimation techniques, hypothesis testing procedures that are resilient to outliers or heavy-tailed distributions, and robust variable selection methods.

First, we consider the problem of robust estimation in partial functional linear models under RKHS framework. The theoretical properties of robust estimation simulation studies are discussed in this chapter. Furthermore, two real data examples are presented to illustrate the performance of the robust procedure.

Then, we extend three robust tests: Wald-type, the likelihood ratio-type and F-type in functional linear models. Meanwhile, we investigate the theoretical properties of these robust testing procedures and assess the finite sample properties through the numerical simulation.

Finally, we propose a robust variable selection method in multiple functional linear regression and present a novel algorithm for identifying significant functional predictors using a robust group variable inflation factor (VIF) selection procedure. Our methodology is validated through rigorous simulation studies as well as its application to real-world data.

To ensure the cohesiveness of this dissertation, Chapter 1 provides an introduction to the research background, mathematical foundations, and primary motivations underlying this study. Chapter 2 presents a comprehensive overview of basis expansion methods for functional data analysis. Lastly, Chapter 6 concludes this dissertation by offering potential avenues for future research.