  Modelling excesses over a high threshold using the Pareto or generalized
Pareto distribution (PD/GPD) is the most popular approach in extreme value
statistics. This method typically requires high thresholds in order for the
(G)PD to fit well and in such a case applies only to a small upper fraction of
the data. The extension of the (G)PD proposed in this paper is able to describe
the excess distribution for lower thresholds in case of heavy tailed
distributions. This yields a statistical model that can be fitted to a larger
portion of the data. Moreover, estimates of tail parameters display stability
for a larger range of thresholds. Our findings are supported by asymptotic
results, simulations and a case study.
