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June 2010 Quantile calculus and censored regression
Yijian Huang
Ann. Statist. 38(3): 1607-1637 (June 2010). DOI: 10.1214/09-AOS771

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.

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Yijian Huang. "Quantile calculus and censored regression." Ann. Statist. 38 (3) 1607 - 1637, June 2010. https://doi.org/10.1214/09-AOS771

Information

Published: June 2010
First available in Project Euclid: 24 March 2010

zbMATH: 1189.62071
MathSciNet: MR2662354
Digital Object Identifier: 10.1214/09-AOS771

Subjects:
Primary: 62N02
Secondary: 62N01

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

Rights: Copyright © 2010 Institute of Mathematical Statistics

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Vol.38 • No. 3 • June 2010
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