Codifference as a measure of dispersion and dependence for mixture models

TL;DR

This paper explores the use of codifference as a versatile measure of dependence and dispersion for various heavy-tailed and non-Gaussian distributions, extending its applicability beyond stable vectors.

Contribution

It introduces a generalized framework for codifference, proposes its natural domain and variants, and demonstrates its effectiveness in analyzing mixture models and detecting non-linear memory.

Findings

Codifference can be applied to a broader class of distributions beyond stable vectors.

It measures bulk properties and ignores tails more than covariance.

The asymptotic distribution of its estimator is derived.

Abstract

Codifference is a commonly used measure of dependence for stable vectors and processes for which covariance is infinite. However, we argue that it can also be used for other heavy-tail distributions and it provides useful information for other non-Gaussian distributions as well, no matter the tails. Motivated by this, we analyse codifference using as little assumptions as possible about the studied model. It leads us to propose its natural domain and three natural variants of it. Using the wide class of variable scale mixture distributions we argue that the codifference can be interpreted as the measure of bulk properties which ignores the tails much more than the covariance. It can also detect forms of non-linear memory which covariance cannot. Finally, we show the asymptotic distribution of its estimator.