Generalised Covariances and Correlations

TL;DR

This paper introduces a generalized framework for covariance and correlation measures using various statistical functionals, providing new dependence metrics with desirable properties and applications to demographic data.

Contribution

The authors develop a novel class of dependence measures based on generalized errors and a Fréchet-Hoeffding normalization, extending traditional covariance and correlation concepts.

Findings

Entire interval [-1, 1] attainable for any marginals

Distributional correlations capture full dependence structure

Application to demographic data demonstrates usefulness

Abstract

The covariance of two random variables measures the average joint deviations from their respective means. We generalise this well-known measure by replacing the means with other statistical functionals such as quantiles, expectiles, or thresholds. Deviations from these functionals are defined via generalised errors, often induced by identification or moment functions. As a normalised measure of dependence, a generalised correlation is constructed. Replacing the common Cauchy-Schwarz normalisation by a novel Fr\'echet-Hoeffding normalisation, we obtain attainability of the entire interval $[-1, 1]$ for any given marginals. We uncover favourable properties of these new dependence measures. The families of quantile and threshold correlations give rise to function-valued distributional correlations, exhibiting the entire dependence structure. They lead to tail correlations, which should arguably supersede the coefficients of tail dependence. Finally, we construct summary covariances (correlations), which arise as (normalised) weighted averages of distributional covariances. We retrieve Pearson covariance and Spearman correlation as special cases. The applicability and usefulness of our new dependence measures is illustrated on demographic data from the Panel Study of Income Dynamics.