  In several applications, ultimately at the largest data, truncation effects
can be observed when analysing tail characteristics of statistical
distributions. In some cases truncation effects are forecasted through physical
models such as the Gutenberg-Richter relation in geophysics, while at other
instances the nature of the measurement process itself may cause under recovery
of large values, for instance due to flooding in river discharge readings.
Recently Beirlant et al. (2016) discussed tail fitting for truncated
Pareto-type distributions. Using examples from earthquake analysis, hydrology
and diamond valuation we demonstrate the need for a unified treatment of
extreme value analysis for truncated heavy and light tails. We generalise the
classical Peaks over Threshold approach for the different max-domains of
attraction with shape parameter $\xi>-1/2$ to allow for truncation effects. We
use a pseudo-maximum likelihood approach to estimate the model parameters and
consider extreme quantile estimation and reconstruction of quantile levels
before truncation whenever appropriate. We report on some simulation
experiments and provide some basic asymptotic results.
