ABSTRACT The multivariate Hilbert-Schmidt Independence Criterion (dHSIC) and distance multivariance allow to measure and test independence of an arbitrary number of random vectors with arbitrary dimensions. Here, we define versions which only depend on an underlying copula. The approach is based on the distributional transform, yielding dependence measures which always feature a natural invariance with respect to scalings and translations. Moreover, it requires no distributional assumptions, i.e., the distributions can be of pure type or any mixture of discrete and continuous distributions and (in our setting) no existence of moments is required. Empirical estimators and tests, which are consistent against all alternatives, are provided based on a Monte Carlo distributional transform. In particular, it is shown that the new estimators inherit the exact limiting distributional properties of the original estimators. Examples illustrate that tests based on the new measures can be more powerful than tests based on other copula dependence measures.

中文翻译：

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通过分布变换的距离多元变量和 dHSIC 的 Copula 版本——构建不变依赖度量的一般方法

摘要 多元希尔伯特-施密特独立准则 (dHSIC) 和距离多元方差允许测量和测试任意维度的任意数量的随机向量的独立性。在这里，我们定义仅依赖于底层 copula 的版本。该方法基于分布变换，产生依赖度量，这些度量总是具有关于缩放和平移的自然不变性。此外，它不需要分布假设，即分布可以是纯类型或离散和连续分布的任何混合，并且（在我们的设置中）不需要存在矩。基于蒙特卡罗分布变换提供与所有备选方案一致的经验估计量和测试。特别是，结果表明，新的估计量继承了原始估计量的精确限制分布特性。示例说明基于新度量的测试可能比基于其他 copula 依赖度量的测试更强大。