Dissertation

Quantile Regression for Censored Data and Spatial Data

Lin, Jilei

2026

Overview

For a random variable with cumulative distribution function (CDF) ( ), the th quantile is defined as ( ) = inf : ( ) ≥ , where 0 < < 1 and inf denotes the infimum of the set . Quantile regression models the conditional quantile function of given covariates x, ( | x). Unlike mean regression, quantile regression is inherently robust to heavy-tailed distributions, outliers, and departures from parametric assumptions, as it characterizes conditional distributional features beyond the mean. Moreover, quantile regression is particularly well suited for settings in which tail heterogeneity is present, allowing covariate effects to vary across different parts of the conditional distribution rather than being summarized by a single average effect. For example, when studying the relationship between years of education and income, conventional mean regression methods such as ordinary least squares may yield estimates similar to those from median regression. However, education may have little association with income among low-income individuals while exerting a much stronger effect in the upper tail of the income distribution, reflecting pronounced heterogeneity between lower and upper quantiles that cannot be captured by mean-based approaches.In biomedical studies, observational data frequently involve early dropout, leading to nonignorable missingness. In such settings, inference based on conditional mean regression is fundamentally limited, since the conditional mean is generally not identifiable without strong parametric assumptions on either the outcome distribution or the missingness mechanism. Beyond capturing tail heterogeneity, quantile regression provides a robust alternative for analyzing missing data, as conditional quantiles remain identifiable for a range of quantile levels under censoring and are therefore less sensitive to missing observations. For many biomedical applications, it is of interest to estimate bent-line regression models, in which the target functional is piecewise linear and continuous across regions defined by unknown change points. However, censoring can induce substantial bias in both slope and change-point estimation for existing mean-based approaches. Motivated by an experimental autoimmune myasthenia gravis (EAMG) dataset, in Chapter 1 we propose a censored bent-line quantile regression (CBQR) framework that formulates the problem directly in terms of identifiable quantile functionals and circumvents parametric assumptions on the missingness mechanism. The resulting estimator is consistent, asymptotically normal, and easy to implement via bent-line quantile regression on an informative subset, with simulation studies and real-data analysis demonstrating substantial bias reduction under nonignorable missingness.In addition to tail heterogeneity, spatial data often exhibit substantial spatial heterogeneity. Existing spatial methods frequently struggle to capture heterogeneous patterns over complex domains or ignore distributional heterogeneity in the tails of the response. In Chapter 2, we introduce a quantile spatial modeling (QSM) framework that accommodates both spatial nonstationarity and tail heterogeneity through constant and spatially varying coefficients. We propose a smoothed quantile bivariate triangulation (SQBiT) method based on penalized splines on triangulations combined with convolution smoothing of the quantile loss. The proposed method effectively captures spatial nonstationarity while preserving important data features such as smoothness and shape over complex and irregular domains. Under mild regularity conditions, the estimator achieves the optimal convergence rate under the 2-norm. We further establish a Bahadur representation, which enables asymptotic normality for the constant coefficient estimator and facilitates construction of asymptotic confidence intervals. To improve finite-sample performance, we also develop a wild bootstrap procedure for inference. Simulation studies demonstrate the numerical and computational advantages of SQBiT over existing methods, and application to U.S. mortality data reveals how socioeconomic factors influence mortality rates differently across spatial regions and distributional tails.Modern spatial datasets are often massive in both volume and resolution, frequently exceeding the memory capacity of a single machine and posing substantial challenges for scalable statistical modeling. While SQBiT performs well for spatial data of moderate size, it does not scale efficiently to large spatial datasets. In Chapter 3, we propose a distributed inference framework for QSM that accommodates both constant and spatially varying coefficients through bivariate triangulation smoothing. The proposed framework is built on a surrogate loss induced by smoothed and decorrelated scores and employs a communication-efficient multi-round aggregation strategy. It achieves the global convergence rate for constant coefficients without requiring large local sample sizes and remains robust even when spatially varying components are imperfectly estimated on local machines. We establish asymptotic normality for the constant coefficient estimator and develop corresponding inference procedures. Simulation studies highlight the numerical performance of the proposed method, and its practical utility is demonstrated through an application to largescale U.S. census-tract–level data on coronary heart disease prevalence.

• Publisher: ProQuest Dissertations & Theses
• Subject: Applied Mathematics / Mathematics / Statistics
• ISBN: 9798247909798