Two-sample Kolmogorov–Smirnov-type tests revisited: Old and new tests in terms of local levels

Abstract

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.