Measuring linear correlation between random vectors

Abstract

We introduce a new scalar coefficient to measure linear correlation between random vectors which preserves all the relevant properties of Pearson’s correlation in arbitrarily large dimensions. The new measure and its bounds are derived from a mass transportation approach in which the expected inner product of two random vectors is taken as a measure of their covariance and then standardized by the maximal attainable value given their marginal covariance matrices. The new correlation is maximized when the average squared Euclidean distance between the random vectors is minimal and attains value one when, additionally, it is possible to establish an affine relationship between the vectors. In several simulative studies we show the limiting distribution of the empirical estimator of the newly defined index and of the corresponding rank correlation.A comparative study based on financial data shows that our proposed correlation, though derived from a novel approach, behaves similarly to some of the multivariate dependence notions recently introduced in the literature.Throughout the paper, we also give some auxiliary results of independent interest in matrix analysis and mass transportation theory, including an improvement to the Cauchy–Schwarz inequality for positive definite covariance matrices.