  This paper develops robust confidence intervals in high-dimensional and
left-censored regression. Type-I censored regression models are extremely
common in practice, where a competing event makes the variable of interest
unobservable. However, techniques developed for entirely observed data do not
directly apply to the censored observations. In this paper, we develop smoothed
estimating equations that augment the de-biasing method, such that the
resulting estimator is adaptive to censoring and is more robust to the
misspecification of the error distribution. We propose a unified class of
robust estimators, including Mallow's, Schweppe's and Hill-Ryan's one-step
estimator.
  In the ultra-high-dimensional setting, where the dimensionality can grow
exponentially with the sample size, we show that as long as the preliminary
estimator converges faster than $n^{-1/4}$, the one-step estimator inherits
asymptotic distribution of fully iterated version. Moreover, we show that the
size of the residuals of the Bahadur representation matches those of the simple
linear models, $s^{3/4 } (\log (p \vee n))^{3/4} / n^{1/4}$ -- that is, the
effects of censoring asymptotically disappear. Simulation studies demonstrate
that our method is adaptive to the censoring level and asymmetry in the error
distribution, and does not lose efficiency when the errors are from symmetric
distributions. Finally, we apply the developed method to a real data set from
the MAQC-II repository that is related to the HIV-1 study.
