Multivariate Gini-type discrepancies

TL;DR

This paper introduces a new scale-invariant discrepancy measure for multivariate distributions, improving upon standard metrics and demonstrated through applications in environmental and social governance analysis.

Contribution

It proposes a novel multivariate Gini-type discrepancy measure that is scale invariant and applicable to multidimensional distributions, with practical advantages over traditional metrics.

Findings

The new discrepancy measure is scale invariant and improves upon mean squared error metrics.

Application to real-world ESG data demonstrates its practical utility.

The measure simplifies analysis of multivariate distributions in various scientific fields.

Abstract

Measuring distances in a multidimensional setting is a challenging problem, which appears in many fields of science and engineering. In this paper, to measure the distance between two multivariate distributions, we introduce a new measure of discrepancy which is scale invariant and which, in the case of two independent copies of the same distribution, and after normalization, coincides with the scaling invariant multidimensional version of the Gini index recently proposed in [34]. A byproduct of the analysis is an easy-to-handle discrepancy metric, obtained by application of the theory to a pair of Gaussian multidimensional densities. The obtained metric does improve the standard metrics, based on the mean squared error, as it is scale invariant. The importance of this theoretical finding is illustrated by means of a real problem that concerns measuring the importance of Environmental, Social and Governance factors for the growth of small and medium enterprises.