In this article, we define extensions of copula-based dependence measures for data with arbitrary distributions, in the non-serial case, i.e., for independent and identically distributed random vectors, as well as in serial case, i.e., for time series. These dependence measures are covariances with respect to a multilinear copula associated with the data. We also consider multivariate extensions based on M\"obius transforms. We find the asymptotic distributions of the statistics under the hypothesis of independence or randomness and under contiguous alternatives. This enables us to find out locally most powerful test statistics for some alternatives, whatever the margins. Numerical experiments are performed for combinations of these statistics to assess the finite sample performance.

翻译：本文定义了在非序列情形（即独立同分布随机向量）与序列情形（即时间序列）下，针对具有任意分布数据的基于Copula的相依性度量扩展。这些相依性度量是与数据相关联的多线性Copula的协方差。我们还考虑了基于Möbius变换的多元扩展。在独立性或随机性假设以及邻近备择假设下，我们找到了这些统计量的渐近分布。这使我们能够针对某些备择假设，在边缘分布未知的情况下，得到局部最优势检验统计量。通过数值实验评估了这些统计量组合的有限样本性能。