Open Access
December 2018 Two-sample Kolmogorov–Smirnov-type tests revisited: Old and new tests in terms of local levels
Helmut Finner, Veronika Gontscharuk
Ann. Statist. 46(6A): 3014-3037 (December 2018). DOI: 10.1214/17-AOS1647


From a multiple testing viewpoint, Kolmogorov–Smirnov (KS)-type tests are union-intersection tests which can be redefined in terms of local levels. The local level perspective offers a new viewpoint on ranges of sensitivity of KS-type tests and the design of new tests. We study the finite and asymptotic local level behavior of weighted KS tests which are either tail, intermediate or central sensitive. Furthermore, we provide new tests with approximately equal local levels and prove that the asymptotics of such tests with sample sizes $m$ and $n$ coincides with the asymptotics of one-sample higher criticism tests with sample size $\min (m,n)$. We compare the overall power of various tests and introduce local powers that are in line with local levels. Finally, suitably parameterized local level shape functions can be used to design new tests. We illustrate how to combine tests with different sensitivity in terms of local levels.


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Helmut Finner. Veronika Gontscharuk. "Two-sample Kolmogorov–Smirnov-type tests revisited: Old and new tests in terms of local levels." Ann. Statist. 46 (6A) 3014 - 3037, December 2018.


Received: 1 May 2016; Revised: 1 September 2017; Published: December 2018
First available in Project Euclid: 7 September 2018

zbMATH: 06968607
MathSciNet: MR3851763
Digital Object Identifier: 10.1214/17-AOS1647

Primary: 62G10 , 62G30
Secondary: 60F99 , 62G20

Keywords: Goodness-of-fit , higher criticism test , local levels , Multiple hypotheses testing , nonparametric two-sample tests , order statistics , weighted Brownian bridge

Rights: Copyright © 2018 Institute of Mathematical Statistics

Vol.46 • No. 6A • December 2018
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