Tail fitting for truncated and non-truncated Pareto-type distributions

Abstract: Recently some papers, such as Aban, Meerschaert and Panorska (2006), Nuyts (2010) and Clark (2013), have drawn attention to possible truncation in Pareto tail modelling. Sometimes natural upper bounds exist that truncate the probability tail, such as the Maximum Possible Loss in insurance treaties. At other instances ultimately at the largest data, deviations from a Pareto tail behaviour become apparent. This matter is especially important when extrapolation outside the sample is required. Given that in practice one does not always know whether the distribution is truncated or not, we consider estimators for extreme quantiles both under truncated and non-truncated Pareto-type distributions. Hereby we make use of the estimator of the tail index for the truncated Pareto distribution first proposed in Aban {\it et al.} (2006). We also propose a truncated Pareto QQ-plot and a formal test for truncation in order to help deciding between a truncated and a non-truncated case. In this way we enlarge the possibilities of extreme value modelling using Pareto tails, offering an alternative scenario by adding a truncation point $T$ that is large with respect to the available data. In the mathematical modelling we hence let $T \to \infty$ at different speeds compared to the limiting fraction ($k/n \to 0$) of data used in the extreme value estimation. This work is motivated using practical examples from different fields of applications, simulation results, and some asymptotic results.