There is the DKW inequality which controls the extent to which the empirical cdf of a sample from a real-valued random variable differs from the true cdf. Are there any stronger results (i.e. equalities, or stronger inequalities) which control the probability addressed by DKW?
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The DKW equality checks the extent of the largest deviation at any point of the ecdf to the cdf. Instead, you could look at the total deviation squared instead (i.e. look at the whole mismatch, instead of the mismatch at a single point). This class of methods has a number of results that typically focus on hypothesis testing such as Anderson-Darling test and the Cramér–von Mises criterion. The hypothesis test equivalent to DKW is the Kolmogorov-Smirnoff test (DKW is essentially an inversion of this test). Papers show that the Anderson-Darling test is a stronger result than the DKW/KS result.
2 Sample Kolmogorov-Smirnov vs. Anderson-Darling vs Cramer-von-Mises
