QUANTILE REGRESSION METHODS IN FUNCTIONAL DATA ANALYSIS

Abstract

In this thesis, we study the partial quantile regression methods in functional data analysis. In the first part, we propose a prediction procedure for the functional linear quantile regression model by using partial quantile covariance techniques and develop a simple partial quantile regression (SIMPQR) algorithm to efficiently extract partial quantile regression (PQR) basis for estimating functional coefficients. In the second part, we propose and implement an alternative formulation of partial quantile regression (APQR) for functional linear model by using block relaxation ideas and finite smoothing techniques. Such reformulation leads to insightful results and motivates new theory, demonstrating consistency and establishing convergence rates by applying advanced techniques from empirical process theory. In the third part, we propose and implement the generalization of PQR procedure to multidimensional functional linear model using tensor decomposition techniques. We also establish and demonstrate the corresponding asymptotic properties. In all three parts, extensive simulations and real data are investigated to show the superiority of our proposed methods, while the advantages of our proposed PQR basis are well demonstrated in various settings for functional linear quantile regression model.