Some Concepts of Dependence
Some Concepts of Dependence
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DOI：
10.1007/978-1-4614-1412-4_64
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发表时间：
1966-10
影响因子：
--
通讯作者：
E. Lehmann
中科院分区：
文献类型：
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作者：
E. Lehmann
摘要和介绍涉及相依变量对（X，Y）的问题在二元正态分布和2 × 2表的情况下得到了最深入的研究。这主要是由于这些事例的重要性，但也许部分地也是由于它们只表现出一种特别简单的依赖形式。(See第7节中的实施例9（i）和10。）涉及一般案例的研究主要围绕两个问题：（i）独立性的检验;（ii）关联度量的定义和估计。在对这些问题的大多数处理中，隐含着一个在其他情况下也很重要的概念（例如，对某些多重决策程序的性能的评估），即正（或负）依赖或关联的概念。独立性检验，例如基于等级相关、肯德尔Z统计量或正态分数的检验，通常不是综合检验（关于此类检验的讨论，见[4]、[15]和[17]，但其目的是检测相当具体的替代品类型，也就是说，大的Y值往往与大的X值相关，小的Y值往往与小的X值相关（正相关）或负相关的相反情况，其中一个变量的大值往往与另一个变量的小值相关联。类似地，关联的度量通常被设计为测量这种关联的程度。本文的目的是给三个依次更强的正相关的定义，调查他们的后果，通过一些例子，探索每个定义的强度，并给出一些统计应用。
Summary and introductionProblems involving dependent pairs of variables (X, Y) have been studied most intensively in the case of bivariate normal distributions and of 2 × 2 tables. This is due primarily to the importance of these cases but perhaps partly also to the fact that they exhibit only a particularly simple form of dependence. (See Examples 9(i) and 10 in Section 7.) Studies involving the general case center mainly around two problems: (i) tests of independence; (ii) definition and estimation of measures of association. In most treatments of these problems, there occurs implicitly a concept which is of importance also in other contexts (for example, the evaluation of the performance of certain multiple decision procedures), the concept of positive (or negative) dependence or association. Tests of independence, for example those based on rank correlation, Kendall’s Z-statistic, or normal scores, are usually not omnibus tests (for a discussion of such tests see [4], [15] and [17], but designed to detect rather specific types of alternatives, namely those for which large values of Y tend to be associated with large values of X and small values of Y with small values of X (positive dependence) or the opposite case of negative dependence in which large values of one variable tend to be associated with small values of the other. Similarly, measures of association are typically designed to measure the degree of this kind of association. The purpose of the present paper is to give three successively stronger definitions of positive dependence, to investigate their consequences, explore the strength of each definition through a number of examples, and to give some statistical applications.