Extreme value statistics for truncated Pareto-type distributions

Recently attention has been drawn to 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 Pareto tail index of a truncated Pareto distribution with a right truncation point $T$. The Hill (1975) estimator is then obtained by 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 whether the distribution is truncated or not, we discuss estimators for the Pareto index and extreme quantiles both under truncated and non-truncated Pareto-type distributions. We also propose a truncated Pareto QQ-plot in order to help deciding between a truncated and a non-truncated case. 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.