Ordinal pattern dependence as a multivariate dependence measure

In this article, we show that the recently introduced ordinal pattern dependence fits into the axiomatic framework of general multivariate dependence measures, i.e., measures of dependence between two multivariate random objects. Furthermore, we consider multivariate generalizations of established univariate dependence measures like Kendall's $\tau$, Spearman's $\rho$ and Pearson's correlation coefficient. Among these, only multivariate Kendall's $\tau$ proves to take the dynamical dependence of random vectors stemming from multidimensional time series into account. Consequently, the article focuses on a comparison of ordinal pattern dependence and multivariate Kendall's $\tau$ in this context. To this end, limit theorems for multivariate Kendall's $\tau$ are established under the assumption of near-epoch dependent data-generating time series. We analyze how ordinal pattern dependence compares to multivariate Kendall's $\tau$ and Pearson's correlation coefficient on theoretical grounds. Additionally, a simulation study illustrates differences in the kind of dependencies that are revealed by multivariate Kendall's $\tau$ and ordinal pattern dependence.