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Dvoretzky--Kiefer--Wolfowitz Inequalities for the Two-sample Case | | Fan Wei
; Richard M Dudley
; | | Date: |
27 Jul 2011 | | 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
$sqrt{n}sup_x|(F_n-F)(x)|$, then there is a finite constant $C$ such that for
any $M>0$, $Pr(K_n>M) leq Cexp(-2M^2).$ Massart proved that one can take C=2
(DKWM inequality) which is sharp for $F$ continuous. We consider the analogous
Kolmogorov--Smirnov statistic $KS_{m,n}$ for the two-sample case and show that
for $m=n$, the DKW inequality holds with C=2 if and only if $ngeq 458$. For
$n_0leq n<458$ it holds for some $C>2$ depending on $n_0$.
For $m
eq n$, the DKWM inequality fails for the three pairs $(m,n)$ with
$1leq m < nleq 3$. We found by computer search that for $ngeq 4$, the DKWM
inequality always holds for $1leq m< nleq 200$, and further that it holds for
$n=2m$ with $101leq mleq 300$. We conjecture that the DKWM inequality holds
for pairs $mleq n$ with the $457+3 =460$ exceptions mentioned. | | Source: | arXiv, 1107.5356 | | Services: | Forum | Review | PDF | Favorites | |
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