Two-sample tests are probably the most commonly used tests in statistics. These tests generally address one aspect of the samples' distribution, such as mean or variance. When the null hypothesis is that two distributions are equal, the Anderson–Darling (AD) test, which is developed from the Cramer–von Mises (CvM) test, is generally employed. Unfortunately, we find that the AD test often fails to identify true differences when the differences are complex: they are not only in terms of mean, variance and/or skewness but also in terms of multi-modality. In such cases, we find that Cramer test, a modification of the CvM test, performs well. However, the adaptation of the Cramer test in routine analysis is hindered by the fact that the mean, variance and skewness of the test statistic are not available, which resulted in the problem of calculating the associated p-value. For this purpose, we propose a new method for obtaining a p-value by approximating the distribution of the test statistic by a generalized Pareto distribution. By approximating the distribution in this way, the calculation of the p-value is much faster than e.g. bootstrap method, especially for large n. We have observed that this approximation enables the Cramer test to have proper control of type-I error. A simulation study indicates that the Cramer test is as powerful as other tests in simple cases and more powerful in more complicated cases.

中文翻译：

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Cramer 检验的属性和近似 p 值计算

双样本检验可能是统计学中最常用的检验。这些测试通常解决样本分布的一个方面，例如均值或方差。当原假设是两个分布相等时，通常采用从 Cramer-von Mises (CvM) 检验发展而来的 Anderson-Darling (AD) 检验。不幸的是，我们发现当差异很复杂时，AD 测试往往无法识别真正的差异：它们不仅在均值、方差和/或偏度方面，而且在多模态方面。在这种情况下，我们发现 Cramer 测试（CvM 测试的修改）表现良好。但是，由于无法获得检验统计量的均值、方差和偏度，因此阻碍了 Cramer 检验在常规分析中的适应性，这导致了计算相关 p 值的问题。为此，我们提出了一种通过广义帕累托分布逼近检验统计量分布来获得 p 值的新方法。通过以这种方式逼近分布，p 值的计算比 bootstrap 方法快得多，尤其是对于大 n。我们已经观察到这种近似使 Cramer 测试能够正确控制 I 类错误。模拟研究表明，Cramer 测试在简单情况下与其他测试一样强大，在更复杂的情况下更强大。p 值的计算比 bootstrap 方法快得多，尤其是对于大 n。我们已经观察到这种近似使 Cramer 测试能够正确控制 I 类错误。模拟研究表明，Cramer 测试在简单情况下与其他测试一样强大，在更复杂的情况下更强大。p 值的计算比 bootstrap 方法快得多，尤其是对于大 n。我们已经观察到这种近似使 Cramer 测试能够正确控制 I 类错误。模拟研究表明，Cramer 测试在简单情况下与其他测试一样强大，在更复杂的情况下更强大。