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 \(\tau\) and Gini's \(\gamma\)) 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 \(\rho\) and a rearranged Kendall's \(\tau\) 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.