A short proof of the Dvoretzky--Kiefer--Wolfowitz--Massart inequality
The Dvoretzky--Kiefer--Wolfowitz--Massart inequality gives a sub-Gaussian tail bound on the supremum norm distance between the empirical distribution function of a random sample and its population counterpart. We provide a short proof of a result that improves the existing bound in two respects. Fir...
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Zusammenfassung: | The Dvoretzky--Kiefer--Wolfowitz--Massart inequality gives a sub-Gaussian
tail bound on the supremum norm distance between the empirical distribution
function of a random sample and its population counterpart. We provide a short
proof of a result that improves the existing bound in two respects. First, our
one-sided bound holds without any restrictions on the failure probability,
thereby verifying a conjecture of Birnbaum and McCarty (1958). Second, it is
local in the sense that it holds uniformly over sub-intervals of the real line
with an error rate that adapts to the behaviour of the population distribution
function on the interval. |
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DOI: | 10.48550/arxiv.2403.16651 |