Jonathan Ansari
University of Salzburg
Room: M3 3127
An ordering for the strength of functional dependence
Classical measures of association, such as Pearson, Spearman or Kendall correlation, quantify the degree of positive/negative (linear) dependence between two random variables. They are increasing with respect to the concordance order that ranges from countermonotonicity to comonotonicity and can be verified for many models.
Recent research in statistics has established new dependence measures that attain values in [0, 1] where 0 characterizes independence of X and Y, and 1 characterizes perfect functional (not necessarily monotone) dependence of Y on X. Examples are Chatterjee's rank correlation, Wasserstein correlations, and rearranged dependence measures. As a natural dependence order underlying such functionals, we introduce the conditional convex order, which ranges from independence to perfect functional dependence. We verify it in settings such as additive error models, the multivariate normal distribution, and various copula-based models, and we give applications to optimal transport. Consequently, our results offer a unified perspective on the behavior of dependence measures across statistical models.