The Cramér–Von Mises Statistic for Continuous Distributions: A Monte Carlo Study for Calculating Its Associated Probability

The Cramér–von Mises (CM) statistic can assess the goodness-of-fit when the null distribution is independent of the underlying distribution, provided it is continuous, resulting in a universality that supports symmetry in its application and interpretation. Along with other order statistics, it serves in a battery of tests. A key element in its use is availability of the cumulative distribution function inverse for calculating the p-value. CM does not have an explicit formula for the cumulative distribution function. Here, a Monte–Carlo experiment was deployed to generate a large amount of data resembling CM. Furthermore, regression analysis was deployed, in order to obtain the dependence of the p-value as function of the statistic and of the sample size. For two cases, one assuming Beta distribution and the other assuming Cauchy distribution, an analysis using the CM statistic was conducted.