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Two-sample Dvoretzky–Kiefer–Wolfowitz inequalities

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  • Wei, Fan
  • Dudley, Richard M.

Abstract

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 nsupx|(Fn−F)(x)|, then there is a constant C such that for any M>0, Pr(Kn>M)≤Cexp(−2M2). 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≥n0 for some C depending on n0, with C=2 if and only if n0≥458.

Suggested Citation

  • Wei, Fan & Dudley, Richard M., 2012. "Two-sample Dvoretzky–Kiefer–Wolfowitz inequalities," Statistics & Probability Letters, Elsevier, vol. 82(3), pages 636-644.
    • Handle: RePEc:eee:stapro:v:82:y:2012:i:3:p:636-644
    DOI: 10.1016/j.spl.2011.11.012
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    Cited by:

    1. Ryan Cumings-Menon, 2022. "Differentially Private Estimation via Statistical Depth," Papers 2207.12602, arXiv.org.
    2. Greene, Evan & Wellner, Jon A., 2016. "Finite sampling inequalities: An application to two-sample Kolmogorov–Smirnov statistics," Stochastic Processes and their Applications, Elsevier, vol. 126(12), pages 3701-3715.

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