Abstract
We consider quantile regression processes from censored data under dependent data structures and derive a uniform Bahadur representation for those processes. We also consider cases where the dimension of the parameter in the quantile regression model is large. It is demonstrated that traditional penalization methods such as the adaptive lasso yield sub-optimal rates if the coefficients of the quantile regression cross zero. New penalization techniques are introduced which are able to deal with specific problems of censored data and yield estimates with an optimal rate. In contrast to most of the literature, the asymptotic analysis does not require the assumption of independent observations, but is based on rather weak assumptions, which are satisfied for many kinds of dependent data.
Highlights
Quantile regression for censored data has found considerable attention in the recent literature
Similar ideas were considered by Bang and Tsiatis (2002) and later Zhou (2006). All these papers have in common that the statistical analysis requires the independence of the censoring times and covariates
One major purpose of the present paper is to demonstrate that a sensible asymptotic theory can be obtained under rather weak assumptions on certain empirical processes that are satisfied for many kinds of dependent data
Summary
Quantile regression for censored data has found considerable attention in the recent literature. Shows et al (2010) proposed to penalize the estimator developed in Zhou (2006) by an adaptive lasso penalty These authors assumed unconditional independence between survival and censoring times and considered only the median. It might not be reasonable to exclude covariates from the model just because they have zero influence at a fixed given quantile All those considerations demonstrate the need for penalization techniques that take into account the special features of quantile regression. The second purpose of the present paper is to construct novel penalization techniques that are flexible enough to deal with the particular properties of censored quantile regression, and to provide a rigorous analysis of the resulting quantile regression processes.
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