Dependence is undoubtedly a central concept in statistics. Though, it proves difficult to locate in the literature a formal definition which goes beyond the self-evident 'dependence = non-independence'. This absence has allowed the term 'dependence' and its declination to be used vaguely and indiscriminately for qualifying a variety of disparate notions, leading to numerous incongruities. For example, the classical Pearson's, Spearman's or Kendall's correlations are widely regarded as 'dependence measures' of major interest, in spite of returning 0 in some cases of deterministic relationships between the variables at play, evidently not measuring dependence at all. Arguing that research on such a fundamental topic would benefit from a slightly more rigid framework, this paper suggests a general definition of the dependence between two random variables defined on the same probability space. Natural enough for aligning with intuition, that definition is still sufficiently precise for allowing unequivocal identification of a 'universal' representation of the dependence structure of any bivariate distribution. Links between this representation and familiar concepts are highlighted, and ultimately, the idea of a dependence measure based on that universal representation is explored and shown to satisfy Renyi's postulates.
翻译:依赖无疑是统计中的一个核心概念。 虽然在文献中很难找到超越不言自明的“ 依赖” = 非独立” 的正式定义。 这种缺失使得“依赖”一词及其偏向性能够被模糊和不加区分地用于限定各种不同的不同概念,导致许多不一致。 例如,古典皮尔逊、斯皮尔曼或肯德尔的关联被广泛视为重大利益的“依赖性衡量标准 ”, 尽管在一些情况中,正在起作用的变量之间有决定性关系,明显没有衡量任何依赖性。 本文认为,对这样一个基本专题的研究将受益于一个略为僵硬的框架, 这表明对同一概率空间上界定的两个随机变量之间的依赖性作了一般性定义。 与直觉一致的自然定义仍然足够精确,足以明确确定任何双轨分布的依赖性结构的“ 普遍性” 代表性。 这种代表性和熟悉的概念之间的联系得到了强调, 最终, 以这一普遍代表性为基础的依赖性衡量概念的概念得到了探讨并显示, 满足了伦伊的后期。</s>