I am wondering whether Dvoretzky–Kiefer–Wolfowitz inequality holds for discrete distributions? Any comments or references would be greatly appreciated.
$\begingroup$
$\endgroup$
1
-
2$\begingroup$ What part of the statement of the inequality is raising doubts as to its validity in the discrete case? $\endgroup$– cardinalCommented Jun 1, 2015 at 17:19
Add a comment
|
1 Answer
$\begingroup$
$\endgroup$
0
Yes, it does. See Theorem 11.6 in Kosorok (2008) Introduction to Empirical Processes and Semiparametric Inference:
Theorem: for any iid sample $X_1,\dots,X_n$ with distribution $F$, $$\mathrm{P}\left(\sup_{t\in\mathbb{R}} \sqrt{n}\, \vert \mathbb{F}_n(t) - F(t)\vert > x\right) \leq 2 \exp(-2x^2),$$ for all $x>0$.
