  This paper considers estimation and inference about tail features when the
observations beyond some threshold are censored. We first show that ignoring
such tail censoring could lead to substantial bias and size distortion, even if
the censored probability is tiny. Second, we propose a new maximum likelihood
estimator (MLE) based on the Pareto tail approximation and derive its
asymptotic properties. Third, we provide a small sample modification to the MLE
by resorting to Extreme Value theory. The MLE with this modification delivers
excellent small sample performance, as shown by Monte Carlo simulations. We
illustrate its empirical relevance by estimating (i) the tail index and the
extreme quantiles of the US individual earnings with the Current Population
Survey dataset and (ii) the tail index of the distribution of macroeconomic
disasters and the coefficient of risk aversion using the dataset collected by
Barro and Urs{\'u}a (2008). Our new empirical findings are substantially
different from the existing literature.
