Convexity and measures of statistical association

原文摘要

Abstract Recent investigations on the measures of statistical association highlight essential properties such as zero-independence (the measure is zero if and only if the random variables are independent), monotonicity under information refinement, and max-functionality (the measure of association is maximal if and only if we are in the presence of a deterministic (noiseless) dependence). An open question concerns the reasons why measures of statistical associations satisfy one or more of those properties but not others. We show that convexity plays a central role in all properties. Convexity plus a form of strictness (that we are to define) are necessary and sufficient for zero-independence, and convexity and strict convexity on Dirac masses are necessary and sufficient for max-functionality. We apply the findings to study the families of measures of statistical association based on Csiszár divergences, optimal transport, kernels, as well as Chatterjee’s new correlation coefficient. We further discuss the role of convexity in guaranteeing the asymptotic unbiasedness of given data estimators, prove a central limit theorem for those estimators under independence, and show the rate of convergence under arbitrary dependence. We demonstrate the findings with numerical simulations in a multivariate response context.

摘要来源：Crossref