Envelopes for censored quantile regression

We propose an efficient estimator for the coefficients in censored quantile regression using the envelope model. The envelope model uses dimension reduction techniques to identify material and immaterial components in the data, and forms the estimator based only on the material component, thus reducing the variability of estimation. We will demonstrate the guaranteed asymptotic efficiency gain of our proposed envelope estimator over the traditional estimator for censored quantile regression. Our analysis begins with the local weighing approach that traditionally relies on semiparametric Z$$ Z $$‐estimation involving the conditional Kaplan–Meier estimator. We will instead invoke the independent identically distributed (i.i.d.) representation of the Kaplan–Meier estimator, which eliminates this infinite‐dimensional nuisance and transforms our objective function in Z$$ Z $$‐estimation into a U$$ U $$‐process indexed by only an Euclidean parameter. The modified Z$$ Z $$‐estimation problem becomes entirely parametric and hence more amenable to analysis. We will also reconsider the i.i.d. representation of the conditional Kaplan–Meier estimator.