Statistical Learning and Inference For Functional Predictor Models via Reproducing Kernel Hilbert Space

Abstract

Functional regression is a cornerstone for understanding complex relationships where predictors or responses (or both) are functions. A particularly powerful framework within this domain is the Reproducing Kernel Hilbert Space (RKHS), which facilitates the handling of infinite-dimensional data through a finite set of parameters.

This thesis delves into three specific topics within functional regression using RKHS, showcasing innovative methodologies and their applications to real-world data. The first topic explores functional linear expectile regression, a method that offers a nuanced view of conditional response distributions, particularly beneficial for asymmetric distributions or when tail behaviour is of interest. The second topic ventures into functional smoothed score (SS) classification. The study investigates the functional classifier’s generalization ability and convergence property. The last chapter addresses the challenge of estimation and inference for the slope function in logistic regression under case-control designs, which exert influence over rare disease research.

The three topics contribute to functional regression, offering robust and theoretically sound methodologies for analyzing complex data structures. Through the representation theorem of RKHS, this thesis advances statistical modeling techniques and provides practical tools for tackling real-world problems in diverse scientific domains.