Finite sampling inequalities: an application to two-sample Kolmogorov-Smirnov statistics

TL;DR

This paper reviews finite-sampling exponential bounds and introduces new results for one-sided two-sample Kolmogorov-Smirnov statistics, extending existing bounds and emphasizing adjusted inequalities.

Contribution

It provides novel exponential bounds for one-sided two-sample Kolmogorov-Smirnov statistics, complementing recent two-sided results and focusing on adjusted inequalities.

Findings

New exponential bounds for one-sided two-sample KS statistics

Extension of Dvoretzky-Kiefer-Wolfowitz and Massart inequalities

Enhanced understanding of finite-sample inequalities in empirical processes

Abstract

We review a finite-sampling exponential bound due to Serfling and discuss related exponential bounds for the hypergeometric distribution. We then discuss how such bounds motivate some new results for two-sample empirical processes. Our development complements recent results by Wei and Dudley (2011) concerning exponential bounds for two-sided Kolmogorov - Smirnov statistics by giving corresponding results for one-sided statistics with emphasis on "adjusted" inequalities of the type proved originally by Dvoretzky, Kiefer, and Wolfowitz (1956) and by Massart (1990) for one-sample versions of these statistics.