Let T be a dependent variable, such as a survival time, and let x be a p 1 covariate vector. Quantile regression methods focus on modeling the conditional quantile function, , which is defined as

For example, is the conditional median quantile, and is the conditional quantile function that corresponds to the 95th percentile.
A linear quantile regression model for has the form . One of the advantages of quantile regression analysis is that the covariate effect can change with . Unlike ordinary least squares regression, which estimates the conditional expectation function , quantile regression offers the flexibility to model the entire conditional distribution.
Given observations , standard quantile regression estimates the regression coefficients by minimizing the following objective function over b:

where
However, in many applications, the responses are subject to censoring. For example, in a biomedical study, censoring occurs when patients withdraw from the study or die from a cause that is unrelated to the disease being studied.
Let denote the censoring variable. In the case of rightcensoring, the triples are observed, where and are the observed response variable and the censoring indicator, respectively. Standard quantile regression leads to a biased estimator of the regression parameters .
The following sections describe two methods for estimating the quantile coefficient in the presence of rightcensoring.