The Cox proportional hazards model and the accelerated failure time model are frequently used in survival data analysis. They are powerful, yet have limitation due to their model assumptions. Quantile regression offers a semiparametric approach to model data with possible heterogeneity. It is particularly powerful for censored responses, where the conditional mean functions are unidentifiable without strict parametric assumptions on the distributions. Unlike the Cox proportional hazards model, quantile regression models the survival times directly, which provides an easy way to explain and extract survival times after the regression parameters are estimated. Recent work by Portnoy (2003) and by Peng and Huang (2008) demonstrated how the Kaplan-Meier estimator and the Nelson-Aaron estimator for univariate samples can be generalized for estimating the conditional quantile functions with right censored data. In this dissertation, we propose a new algorithm to compute regression quantiles when the response variable is subject to double censoring. In this area, little has been done. The proposed algorithm distributes the probability mass of each censored point to its left or right in a self-consistent manner, taking the idea of Turnbull (1974) to a broader platform. The algorithm is insensitive to starting values, and can be used to estimate a set of quantile functions with a small number of iterations. The large sample properties of the estimator are studied. Numerical results on simulated data and empirical data are carried out to demonstrate the merits of the proposed method. Extension to more general forms of censoring is also explored. [The dissertation citations contained here are published with the permission of ProQuest LLC. Further reproduction is prohibited without permission. Copies of dissertations may be obtained by Telephone (800) 1-800-521-0600. Web page: http://www.proquest.com/en-US/products/dissertations/individuals.shtml.]