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Goodness-of-fit, correlation tests, Cramér-von Mises, empirical and quantile/processes, Kolmogorov-Smirnov, Shapiro-Wilk, strong approximations, Wasserstein distance, Primary 62F05, 62E20, secondary 60F25,

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SUMMARY It is shown that an asymptotically precise one‐term correction to the asymptotic distribution function of the classical Cramér‐von Mises statistic approximates the exact distribution function remarkably closely for sample sizes as small as 7 or even smaller. This correction can be quickly evaluated, and hence it is suitable for the computation of practically exact p‐values when testing simple goodness of fit. Similar findings hold for Watson's rotationally invariant modification, where a sample size of 4 appears to suffice.

No abstract