Convexity and measures of statistical association
研究了凸性在统计关联度量性质（如零独立性、信息细化单调性和最大功能性）中的核心作用，并应用于Csiszár散度、最优传输、核方法及Chatterjee相关系数等度量族，还讨论了凸性对估计量渐近无偏性和中心极限定理的影响。
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.