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Can the Dvoretzky–Kiefer–Wolfowitz (a.k.a. Massart) inequality

$$ \Pr\left( \sup |\hat F_n(x) - F(x)| > \lambda \right) \le 2\exp(-2n\lambda^2) $$

where $\hat F_n(x)$ is the empirical distribution function of the sample $X$ from the distribution $F$, be extended to comparing two empirical distributions coming $\hat F_n,\hat G_n$ coming from equally-sized samples $X,Y$ from the same distribution? What is the $ \Pr\left( \sup |\hat F_n(x) - \hat G_n(x)| > \lambda \right) $ ?

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  • $\begingroup$ Maybe it's too early for me but can't we deduce from $sup|\hat{F}_n(x) - F(x)| < \lambda$ and $sup|\hat{G}_n(x) - F(x)| < \lambda$ that $$sup|\hat{F}_n(x) - \hat{G}_n(x)| < 2\lambda$$ $\endgroup$
    – Łukasz Grad
    Commented Apr 18, 2017 at 9:34
  • $\begingroup$ Adding the maximum errors together works, but is too conservative. $\endgroup$
    – Bscan
    Commented Apr 16, 2018 at 21:00

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Yes, the DKW inequality is essentially an inversion of the one sample Kolmogorov Smirnoff test. A two-sample version exists for it too where the primary difference is the scaling of the constant. So yes, you could invert this test too.

$$ D>c(\alpha)\sqrt{\frac{n + m}{n m}} $$

Here's a paper describing the process in detail: https://arxiv.org/pdf/1107.5356.pdf

https://en.wikipedia.org/wiki/Kolmogorov%E2%80%93Smirnov_test#Two-sample_Kolmogorov%E2%80%93Smirnov_test

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