DVORETZKY–KIEFER–WOLFOWITZ INEQUALITIES FOR THE TWO-SAMPLE CASE Citation
نویسندگان
چکیده
The Dvoretzky–Kiefer–Wolfowitz (DKW) inequality says that if Fn is an empirical distribution function for variables i.i.d. with a distribution function F , and Kn is the Kolmogorov statistic √ n supx |(Fn − F )(x)|, then there is a finite constant C such that for any M > 0, Pr(Kn > M) ≤ C exp(−2M2). Massart proved that one can take C = 2 (DKWM inequality) which is sharp for F continuous. We consider the analogous Kolmogorov– Smirnov statistic KSm,n for the two-sample case and show that for m = n, the DKW inequality holds with C = 2 if and only if n ≥ 458. For n0 ≤ n < 458 it holds for some C > 2 depending on n0. For m 6= n, the DKWM inequality fails for the three pairs (m, n) with 1 ≤ m < n ≤ 3. We found by computer search that for n ≥ 4, the DKWM inequality always holds for 1 ≤ m < n ≤ 200, and further that it holds for n = 2m with 101 ≤ m ≤ 300. We conjecture that the DKWM inequality holds for pairs m ≤ n with the 457 + 3 = 460 exceptions mentioned.
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