  Most extreme events in real life can be faithfully modeled as random
realizations from a Generalized Pareto distribution, which depends on two
parameters: the scale and the shape. In many actual situations, one is mostly
concerned with the shape parameter, also called tail index, as it contains the
main information on the likelihood of extreme events. In this paper, we show
that the mean/max statistic, that is the empirical mean divided by the maximal
value of the sample, constitutes an ideal normalization to study the tail index
independently of the scale. This statistic appears naturally when trying to
distinguish between uniform and exponential distributions, the two transitional
phases of the Generalized Pareto model. We propose a simple methodology based
on the mean/max statistic to detect, classify and infer on the tail of the
distribution of a sample. Applications to seismic events and detection of
saturation in experimental measurements are presented.
