Estimation of the Maximal Moment Exponent with Censored Data

Heavy-tailed distributions have been used tomodel phenomena in which extreme events occur with high probability. In these type of occurrences, it is likely that extreme events are not observable after a certain threshold. Appropriate estimators are needed to deal with this type of censored data. We show that the well-known Hill-Hall estimator is unable to deal with censored data and yields highly biased estimates. We propose and study an unbiased modified maximum likelihood estimator, as well as a truncated tail regression estimator. We assess the expected value and the variance of these estimators in the cases of stableand Pareto-distributed data.