Invariance properties of dependence measures

Abstract

Li gave a generalization of non-symmetric copula-based dependence measure, such as the Trutschnig\'s measure of dependence. A precise sufficient condition which makes Li\'s generalization a non-symmetric measure of dependence is given and proved rigorously. Supported by its non-symmetric dependence measure properties, we symmetrize the Li\'s non-symmetric measure of dependence and investigate its properties. Specifically, we analyze the key properties of dependence measures including well-defined property, abilities to detect independence and dependence at the two extreme values $0,1$ respectively, and invariance under the certain types of transformations. In particular, we find, via several examples, that a dependence measure possessing an ability to detect a larger class of dependences tends to be invariant under an accordingly large class of transformations. The probabilistic version of maximal information coefficient (MIC) is also proved to be a dependence measure. Lastly, we show that there does not exist a dependence measure which is both invariant under strictly monotonic transformations and able to catch complete dependence.