Extreme value statistics for truncated Pareto-type distributions

Recently some authors have drawn attention to the fact that there might be practical problems with the use of unbounded Pareto distributions, for instance when there are natural upper bounds that truncate the probability tail. Aban, Meerschaert and Panorska (2006) derived the maximum likelihood estimator for the tail index of a truncated Pareto distribution with right truncation point $T$. The Hill (1975) estimator is then obtained from this maximum likelihood estimator letting $T \to \infty$. The problem of extreme value estimation under (right) truncation was also introduced in Nuyts (2010) who proposed a similar estimator for the tail index and considered trimming of the number of extreme order statistics. Given that in practice one does not always know if the distribution is truncated or not, we propose estimators for extreme quantiles and $T$ that are consistent both under truncated and non-truncated Pareto-type distributions. In this way we extend the classical extreme value methodology adding the truncated Pareto-type model with truncation point $T \to \infty$ as the sample size $n \to \infty$. Finally we present some practical examples, asymptotics and simulation results.