Centre-free kurtosis orderings for asymmetric distributions

TL;DR

This paper develops a rigorous framework for comparing the kurtosis of asymmetric distributions by focusing on distributions with equal skewness, introducing a functional that preserves order, and analyzing specific distribution families.

Contribution

It introduces a novel kurtosis ordering for asymmetric distributions that is transitive when skewness is fixed, addressing limitations of previous approaches.

Findings

Kurtosis orderings are non-transitive due to skewness entanglement.

Restricting to equal skewness makes kurtosis ordering transitive.

The sinh-arsinh distribution shows skewness-invariant kurtosis behavior.

Abstract

The concept of kurtosis is used to describe and compare theoretical and empirical distributions in a multitude of applications. In this connection, it is commonly applied to asymmetric distributions. However, there is no rigorous mathematical foundation establishing what is meant by kurtosis of an asymmetric distribution and what is required to measure it properly. All corresponding proposals in the literature centre the comparison with respect to kurtosis around some measure of central location. Since this either disregards critical amounts of information or is too restrictive, we instead revisit a canonical approach that has barely received any attention in the literature. It reveals the non-transitivity of kurtosis orderings due to an intrinsic entanglement of kurtosis and skewness as the underlying problem. This is circumvented by restricting attention to sets of distributions with equal skewness, on which the proposed kurtosis ordering is shown to be transitive. Moreover, we introduce a functional that preserves this order for arbitrary asymmetric distributions. As application, we examine the families of Weibull and sinh-arsinh distributions and show that the latter family exhibits a skewness-invariant kurtosis behaviour.