标题： 名义随机变量的恰当相关系数 显示英文标题

标题： Proper Correlation Coefficients for Nominal Random Variables

摘要： 本文提出了一种直观的二变量完全依赖的概念，其中至少有一个变量具有可以在所有边缘分布中达到的名义尺度，并提出了一组依赖度量，这些度量当且仅当满足这种完全依赖时值为1。 相对于经典的依赖度量（如列联系数、古德曼-克鲁斯卡尔的$\lambda$和$\tau$以及所谓的不确定系数），这些依赖度量的优势有两个方面。 首先，如果其中一个变量是实值并且表现出连续性，则它们可以被定义。 其次，它们满足可达性属性。 也就是说，无论涉及的边缘分布如何，它们都可以在区间[0,1]内取所有值。 这两个性质经典依赖度量并不具备，后者需要两个离散的边缘分布，并且在某些情况下即使依赖关系很强甚至完全依赖时也可能接近于0。 此外，我为其中一个新依赖度量提供了一个一致估计量及其在独立性和一般情况下的渐近分布。 这使得构造置信区间和一个独立性检验成为可能，随后我在一个模拟研究中检查了该检验在有限样本中的表现。 最后，我通过两个应用实例展示了新依赖度量的使用，分别探讨了国家与收入之间以及国家与宗教之间的依赖关系。 显示英文摘要

摘要： This paper develops an intuitive concept of perfect dependence between two variables of which at least one has a nominal scale that is attainable for all marginal distributions and proposes a set of dependence measures that are 1 if and only if this perfect dependence is satisfied. The advantages of these dependence measures relative to classical dependence measures like contingency coefficients, Goodman-Kruskal's lambda and tau and the so-called uncertainty coefficient are twofold. Firstly, they are defined if one of the variables is real-valued and exhibits continuities. Secondly, they satisfy the property of attainability. That is, they can take all values in the interval [0,1] irrespective of the marginals involved. Both properties are not shared by the classical dependence measures which need two discrete marginal distributions and can in some situations yield values close to 0 even though the dependence is strong or even perfect. Additionally, I provide a consistent estimator for one of the new dependence measures together with its asymptotic distribution under independence as well as in the general case. This allows to construct confidence intervals and an independence test, whose finite sample performance I subsequently examine in a simulation study. Finally, I illustrate the use of the new dependence measure in two applications on the dependence between the variables country and income or country and religion, respectively.