The Annals of Statistics

Quantile calculus and censored regression

Yijian Huang

Full-text: Open access

Abstract

Quantile regression has been advocated in survival analysis to assess evolving covariate effects. However, challenges arise when the censoring time is not always observed and may be covariate-dependent, particularly in the presence of continuously-distributed covariates. In spite of several recent advances, existing methods either involve algorithmic complications or impose a probability grid. The former leads to difficulties in the implementation and asymptotics, whereas the latter introduces undesirable grid dependence. To resolve these issues, we develop fundamental and general quantile calculus on cumulative probability scale in this article, upon recognizing that probability and time scales do not always have a one-to-one mapping given a survival distribution. These results give rise to a novel estimation procedure for censored quantile regression, based on estimating integral equations. A numerically reliable and efficient Progressive Localized Minimization (PLMIN) algorithm is proposed for the computation. This procedure reduces exactly to the Kaplan–Meier method in the k-sample problem, and to standard uncensored quantile regression in the absence of censoring. Under regularity conditions, the proposed quantile coefficient estimator is uniformly consistent and converges weakly to a Gaussian process. Simulations show good statistical and algorithmic performance. The proposal is illustrated in the application to a clinical study.

Article information

Source
Ann. Statist. Volume 38, Number 3 (2010), 1607-1637.

Dates
First available in Project Euclid: 24 March 2010

Permanent link to this document
http://projecteuclid.org/euclid.aos/1269452649

Digital Object Identifier
doi:10.1214/09-AOS771

Mathematical Reviews number (MathSciNet)
MR2662354

Zentralblatt MATH identifier
05712433

Subjects
Primary: 62N02: Estimation
Secondary: 62N01: Censored data models

Keywords
Differential equation estimating integral equation piecewise-linear programming PLMIN algorithm quantile equality fraction regression quantile relative quantile varying-coefficient model

Citation

Huang, Yijian. Quantile calculus and censored regression. Ann. Statist. 38 (2010), no. 3, 1607--1637. doi:10.1214/09-AOS771. http://projecteuclid.org/euclid.aos/1269452649.


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