Advances in Semiparametric Methods for Censored Data and Endogeneity

This thesis delves into two semiparametric regression methods: the Proportional Hazards (PH) model and Quantile Regression (QR) within Survival Analysis.

Our initial goal is to create a methodology to compare two independent or paired samples in survival analysis. This encompasses both conditional and unconditional outcomes, using QR as a primary comparison tool. A novel test is introduced to compare conditional quantile curves with right-censored duration data, suitable for both independent and paired samples. We further discuss the performance of these proposed tests, compare them to existing methods, and explore their real-world applications.

The subsequent objective addresses endogeneity in survival analysis for right-censored outcomes within QR and PH models. We enhance the QR model literature by integrating endogenous variables and random right censoring. Assuming linearity between the structural quantile of the outcome variable's logarithm and the covariates, we treat the censoring variable as independent. Our introduced estimation procedure solves an empirical system of equations derived from the model, detailing the conditions where our estimator is asymptotically normal.

For the PH model, we propose an innovative method to navigate the endogeneity challenge using instrumental variables. A novel presmoothing technique is introduced, which first estimates the model nonparametrically. Following this, "proxy" observations under exogeneity based on the nonparametric estimation are produced and subjected to the standard partial likelihood estimator. While the chapter primarily centers on the PH model, our presmoothing strategy can be adapted for other semiparametric instrumental variable QR models.

Lastly, this thesis provides a comprehensive overview of the developed statistical methodologies and discusses potential future expansions.