Invariant Correlation under Marginal Transforms

A useful property of independent samples is that their correlation remains the sameafter applying marginal transforms. This invariance property plays a fundamental role instatistical 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 abivariate 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 preciselydescribed by what we call positive Frechet 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 invariantcorrelation 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.