Fix-k Asymptotically Unbiased Estimation of Tail Properties with Complete, Censored, or Truncated Data

This paper considers estimating tail properties such as high quantiles and tail conditional expectations. We provide new asymptotically (quantile) unbiased estimators that are applicable to (i) complete data; (ii) tail censored (top-coded) data with known or unknown censoring value; and (iii) tail truncated data with known and unknown truncation value. The new method relies on the sole assumption that the largest k observations satisfy the extreme value theory, for a given and fixed k. This asymptotics leads to excellent small sample bias and risk properties as shown by Monte Carlo simulations, and the empirical relevance is illustrated by estimating the high quantiles of the U.S. hurricane damage. In addition to i.i.d. data, the new method is generalized to accommodate stochastic volatility models by proving that the residuals of fitting a correctly specified AR-GARCH model satisfy our assumption.