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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.
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Csörgő et al. (1996) studied this question.
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