The Mean/Max Statistic in Extreme Value Analysis

TL;DR

This paper introduces the mean/max statistic as an effective tool for analyzing the tail index in extreme value distributions, aiding in distinguishing distribution types and inferring tail behavior.

Contribution

It proposes a novel methodology using the mean/max statistic to detect, classify, and infer the tail properties of distributions in extreme value analysis.

Findings

The mean/max statistic effectively distinguishes between uniform and exponential distributions.

Application to seismic data demonstrates practical utility.

Method detects saturation in experimental measurements.

Abstract

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.