Cramér–von Mises CDF Gap
Compare integrated CDF error with a fully specified null instead of a single largest gap.
Free theory articleCompare integrated CDF error with a fully specified null instead of a single largest gap.
Concept & interpretation
The Cramér–von Mises (CvM) statistic evaluates the goodness of fit of a cumulative distribution function (CDF) compared to an empirical CDF. Unlike the Kolmogorov-Smirnov statistic which looks at the single maximum difference, the CvM statistic integrates the squared difference over the entire distribution.
Interactive Experiment
You can interactively compare different distributions by exploring the Cramér–von Mises CDF Gap module.
- Select the Sample "Even" and the Null CDF "Uniform". Notice the statistic is exactly 0.020833333333333332 as derived above.
- Change the Sample to "Clustered". The ECDF visually diverges more significantly from the uniform diagonal, and the statistic increases to 0.5233333333333333, reflecting higher discrepancy.
- Change the Null CDF to "Quadratic" while keeping the "Even" sample. The points no longer align perfectly with the curve F(x) = x2, producing a statistic of 0.15462239583333334.