A novel positive dependence property is introduced, called positive measure inducing (PMI for short), being fulfilled by numerous copula classes, including Gaussian, Fr\'echet, Farlie-Gumbel-Morgenstern and Frank copulas; it is conjectured that even all positive quadrant dependent Archimedean copulas meet this property. From a geometric viewpoint, a PMI copula concentrates more mass near the main diagonal than in the opposite diagonal. A striking feature of PMI copulas is that they impose an ordering on a certain class of copula-induced measures of concordance, the latter originating in Edwards et al. (2004) and including Spearman's rho $\rho$ and Gini's gamma $\gamma$, leading to numerous new inequalities such as $3 \gamma \geq 2 \rho$. The measures of concordance within this class are estimated using (classical) empirical copulas and the intrinsic construction via empirical checkerboard copulas, and the estimators' asymptotic behaviour is determined. Building upon the presented inequalities, asymptotic tests are constructed having the potential of being used for detecting whether the underlying dependence structure of a given sample is PMI, which in turn can be used for excluding certain copula families from model building. The excellent performance of the tests is demonstrated in a simulation study and by means of a real-data example.

翻译：本文引入了一种新颖的正相依性质，称为正度量诱导性（简称PMI），该性质被众多Copula类所满足，包括高斯、弗雷歇、法利-冈贝尔-摩根斯特恩和弗兰克Copula；有猜想认为，甚至所有正象限相依的阿基米德Copula也满足此性质。从几何角度看，PMI Copula在主对角线附近聚集的质量多于反对角线。PMI Copula的一个显著特征是，它们对一类由Copula诱导的和谐度量施加了排序，这类度量源自Edwards等人（2004），包括斯皮尔曼ρ（$\rho$）和基尼γ（$\gamma$），进而得到诸如$3\gamma \geq 2\rho$等众多新的不等式。使用（经典）经验Copula和通过经验棋盘Copula的内在构造方法，对这类和谐度量进行估计，并确定了估计量的渐近行为。基于所提出的不等式，构建了渐近检验，这些检验具有检测给定样本的潜在相依结构是否为PMI的潜力，进而可用于从模型构建中排除某些Copula族。通过仿真研究和实际数据示例，展示了这些检验的优异性能。