Abstract

This work proposes a new method for computing acceptance regions of exact multinomial tests. From this an algorithm is derived, which finds exact p-values for tests of simple multinomial hypotheses. Using concepts from discrete convex analysis, the method is proven to be exact for various popular test statistics, including Pearson’s Chi-square and the log-likelihood ratio. The proposed algorithm improves greatly on the naive approach using full enumeration of the sample space. However, its use is limited to multinomial distributions with a small number of categories, as the runtime grows exponentially in the number of possible outcomes. The method is applied in a simulation study, and uses of multinomial tests in forecast evaluation are outlined. Additionally, properties of a test statistic using probability ordering, referred to as the “exact multinomial test” by some authors, are investigated and discussed. The algorithm is implemented in the accompanying R package ExactMultinom. Supplementary materials for this article are available online.

摘要

这项工作提出了一种计算精确多项式检验的接受区域的新方法。由此衍生出一个算法，该算法可以找到精确的p-简单多项假设检验的值。使用离散凸分析的概念，该方法被证明对于各种流行的检验统计数据是准确的，包括皮尔逊卡方和对数似然比。所提出的算法大大改进了使用样本空间的完整枚举的朴素方法。然而，它的使用仅限于具有少量类别的多项分布，因为运行时间会随着可能结果的数量呈指数增长。该方法应用于模拟研究，并概述了多项式检验在预测评估中的应用。此外，还研究和讨论了使用概率排序的检验统计量的属性，一些作者将其称为“精确多项式检验”。该算法在随附的 R 包中实现精确多项式。本文的补充材料可在线获取。