INCLUSION-EXCLUSION-BONFERRONI IDENTITIES AND INEQUALITIES FOR DISCRETE TUBE-LIKE PROBLEMS VIA EULER CHARACTERISTICS
INCLUSION-EXCLUSION-BONFERRONI IDENTITIES AND INEQUALITIES FOR DISCRETE TUBE-LIKE PROBLEMS VIA EULER CHARACTERISTICS
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基于欧拉特征的离散类管问题的包含-排除-BONFERRONI 恒等式和不等式
DOI：
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发表时间：
1992
期刊：
影响因子：
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通讯作者：
H. Wynn
中科院分区：
文献类型：
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作者：
D. Naiman;H. Wynn
开发了对经典包容性排斥身份的改进。有两个主要结果：抽象的组合结果和一个具体的几何结果。在抽象的结果条件下，可以保证存在深度$ d + 1 $的身份或不平等的标识或不平等，以指示有限事件集合的指示函数，即一种表达式，即是指示函数的线性组合最多$（D + 1）$ - 折叠事件的交集。可以将这种身份或不平等相对于任何概率措施集成，以产生概率身份或不平等。与Bonferroni-Type不平等现象的先前工作有关。具体的结果表明，在$ d $维的欧几里得空间中，有限的许多球有一个深度$ d + 1 $的身份。有了一个校正项，此结果也包含在$ d $维度的领域中。这些结果构成了离散的管子理论的基础，该理论到目前为止一直是连续的。球形结果用于给出一种模拟方法，以查找多个镜头程序的关键概率，并描述了实施该方法的计算机程序。给出了数值结果，该结果表明，在分布概率估计的尾巴中，基于该方法的估计值往往比基于天真模拟的估计值较小。
Improvements to the classical inclusion-exclusion identity are developed. There are two main results: an abstract combinatoric result and a concrete geometric result. In the abstract result conditions are given which guarantee the existence of a depth $d + 1$ identity or inequality for the indicator function of a union of a finite collection of events, that is, an expression which is a linear combination of indicator functions of at most $(d + 1)$-fold intersections of the events. Such an identity or inequality can be integrated with respect to any probability measure to yield a probability identity or inequality. Connections are given to previous work on Bonferroni-type inequalities. The concrete result says that there is a depth $d + 1$ identity for the union of finitely many balls in $d$-dimensional Euclidean space. With a single correction term this result also holds in the $d$-dimensional sphere. These results form the basis for a discrete theory of tubes, which up to now has been continuous in nature. The spherical result is used to give a simulation method for finding critical probabilities for multiple-comparisons procedures, and a computer program implementing the method is described. Numerical results are presented which demonstrate that in the tails of the distribution probability estimates based on the method tend to exhibit less variability than estimates based on naive simulation.