Invariant correlation under marginal transforms

TL;DR

This paper investigates the property of invariant correlation under marginal transforms in dependent samples, characterizing models with this property and exploring their applications in multivariate dependence structures.

Contribution

It provides a comprehensive characterization of models with invariant correlation, including exchangeable copulas and multivariate correlation matrices, and introduces a new positive regression dependent model.

Findings

Characterization of invariant correlation models via comonotonicity and independence

Identification of positive Fréchet copulas as exchangeable copulas with invariant correlation

Development of a positive regression dependent model for prescribed invariant correlation matrices

Abstract

A useful property of independent samples is that their correlation remains the same after applying marginal transforms. This invariance property plays a fundamental role in statistical inference, but does not hold in general for dependent samples. In this paper, we study this invariance property on the Pearson correlation coefficient and its applications. A multivariate random vector is said to have an invariant correlation if its pairwise correlation coefficients remain unchanged under any common marginal transforms. For a bivariate case, we characterize all models of such a random vector via a certain combination of comonotonicity -- the strongest form of positive dependence -- and independence. In particular, we show that the class of exchangeable copulas with invariant correlation is precisely described by what we call positive Fr\'echet copulas. In the general multivariate case, we characterize the set of all invariant correlation matrices via the clique partition polytope. We also propose a positive regression dependent model that admits any prescribed invariant correlation matrix. Finally, we show that all our characterization results of invariant correlation, except one special case, remain the same if the common marginal transforms are confined to the set of increasing ones.