A Pareto tail plot without moment restrictions

TL;DR

This paper introduces a new Pareto tail plot that remains meaningful for all distributions, including those without finite moments, by utilizing a novel mean functional and a tail estimator based on U-statistics.

Contribution

It proposes a new tail plot and mean functional that characterize Pareto distributions without moment restrictions, along with an estimator and its large sample properties.

Findings

The new plot is applicable to distributions without finite mean.

The estimator based on U-statistics performs well in large samples.

Application to loss datasets demonstrates practical usefulness.

Abstract

We propose a mean functional which exists for any probability distributions, and which characterizes the Pareto distribution within the set of distributions with finite left endpoint. This is in sharp contrast to the mean excess plot which is not meaningful for distributions without existing mean, and which has a nonstandard behaviour if the mean is finite, but the second moment does not exist. The construction of the plot is based on the so called principle of a single huge jump, which differentiates between distributions with moderately heavy and super heavy tails. We present an estimator of the tail function based on $U$-statistics and study its large sample properties. The use of the new plot is illustrated by several loss datasets.