## Abstract

The Dvoretzky-Kiefer-Wolfowitz (DKW) inequality says that if F _{n} is an empirical distribution function for variables i.i.d. with a distribution function F, and K _{n} is the Kolmogorov statistic nsupx{pipe}(Fn-F)(x){pipe}, then there is a constant C such that for any M>0, Pr(K _{n}>M)≤Cexp(-2M ^{2}). Massart proved that one can take C=2 (DKWM inequality), which is sharp for F continuous. We consider the analogous Kolmogorov-Smirnov statistic for the two-sample case and show that for m=n, the DKW inequality holds for n≥n _{0} for some C depending on n _{0}, with C=2 if and only if n _{0}≥458.The DKWM inequality fails for the three pairs (m, n) with 1 ≤ m< n≤ 3. We found by computer search that the inequality always holds for n≥ 4 if 1 ≤ m< n≤ 200, and further for n= 2 m if 101 ≤ m≤ 300. We conjecture that the DKWM inequality holds for all pairs m≤ n with the 457 + 3 = 460 exceptions mentioned.

Original language | English (US) |
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Pages (from-to) | 636-644 |

Number of pages | 9 |

Journal | Statistics and Probability Letters |

Volume | 82 |

Issue number | 3 |

DOIs | |

State | Published - Mar 2012 |

## All Science Journal Classification (ASJC) codes

- Statistics and Probability
- Statistics, Probability and Uncertainty

## Keywords

- Empirical distribution functions
- Kolmogorov-Smirnov test