A comparison method for heavy-tailed random variables

TL;DR

This paper introduces a new non-parametric risk measure for classifying heavy tails of random variables, extending classical tail indices and enabling improved tail estimation.

Contribution

It develops a novel, non-parametric risk measure based on tail decay, generalizing classical indices and providing a new framework for tail comparison and classification.

Findings

The risk measure effectively captures tail decay speed.

The method applies to all heavy-tailed distributions.

Numerical examples demonstrate the measure's properties.

Abstract

We investigate a way of comparing and classifying tails of random variables. Our approach extends the notion of classical indices, such as exponential and moment indices, which are widely used measuring heaviness of tail functions. A non-parametric risk measure applicable for all heavy-tailed random variables is obtained as a concave function that represents the decay speed of tail function. Many key properties of the distribution of a random variable are encoded into this function, which enables a new way to estimate tails. The latter half of the paper is devoted to numerous examples illustrating properties of the results developed during the first half.