Rearranged dependence measures

Most of the popular dependence measures for two random variables X and Y (such as Pearson's and Spearman's correlation, Kendall's τ and Gini's γ) vanish whenever X and Y are independent. However, neither does a vanishing dependence measure necessarily imply independence, nor does a measure equal to 1 imply that one variable is a measurable function of the other. Yet, both properties are natural properties for a convincing dependence measure. In this paper, we present a general approach to transforming a given dependence measure into a new one which exactly characterizes independence as well as functional dependence. Our approach uses the concept of monotone rearrangements as introduced by Hardy and Littlewood and is applicable to a broad class of measures. In particular, we are able to define a rearranged Spearman's ρ and a rearranged Kendall's τ which do attain the value 0 if and only if both variables are independent, and the value 1 if and only if one variable is a measurable function of the other. We also present simple estimators for the rearranged dependence measures, prove their consistency and illustrate their finite sample properties by means of a simulation study and a data example.