Distribution free goodness of fit tests for regularly varying tail distributions

TL;DR

This paper introduces a new class of distribution-free goodness-of-fit tests for regularly varying tail distributions, leveraging maximum likelihood estimation to transform tail data into a distribution-free framework.

Contribution

It proposes a novel approach to construct asymptotic distribution-free tests for tail distributions by treating tails as parametric families and using MLE for exponent estimation.

Findings

Demonstrates the asymptotic behavior of new tests

Provides a method to avoid choosing specific tail process functionals

Validates the tests through theoretical analysis

Abstract

We discuss in this paper a possibility of constructing a whole class of asymptotic distribution-free tests for testing regularly varying tail distributions. The idea is that we treat the tails of distributions as members of a parametric family and using MLE to estimate the exponent. No matter what the exponent's estimator is, we are able to transform the whole class into a specific distribution with a prefix exponent so that we are free from choosing any functional of the tail empirical process as a distribution-free test statistic. The asymptotic behavior of some new tests, as examples from the whole class of new tests, are demonstrated as well.