Distance Correlation Coefficients for Lancaster Distributions

TL;DR

This paper develops a series representation for calculating distance correlation coefficients specifically for Lancaster distributions, including normal and certain discrete distributions, enhancing understanding of their dependence measures.

Contribution

It introduces a general series formula for distance covariance in Lancaster distributions and applies it to various well-known distributions, providing explicit expressions.

Findings

Derived a general series representation for distance covariance.

Obtained explicit formulas for normal and Lancaster-type distributions.

Extended the theory to multivariate and discrete distributions.

Abstract

We consider the problem of calculating distance correlation coefficients between random vectors whose joint distributions belong to the class of Lancaster distributions. We derive under mild convergence conditions a general series representation for the distance covariance for these distributions. To illustrate the general theory, we apply the series representation to derive explicit expressions for the distance covariance and distance correlation coefficients for the bivariate normal distribution and its generalizations of Lancaster type, the multivariate normal distributions, and the bivariate gamma, Poisson, and negative binomial distributions which are of Lancaster type.