On preferences of general two-sided tests with applications to Kolmogorov–Smirnov-type tests

Power functions of tests for Gaussian shift experiments on infinite dimensional Hilbert spaces usually can not be calculated explicitly. Therefore one analyzes the behavior of such tests in the neighborhood of the null hypothesis. Useful measures to compare the quality of different testing procedures are the gradient of a one-sided and the curvature of a two-sided test in the null hypothesis. Janssen (1995) showed that a principal component decomposition of the curvature exists based on a Hilbert–Schmidt operator. It follows that these tests have only acceptable power for a finite number of directions. In this paper we prove an even stronger general result for Gauss shifts under just mild additional assumptions. A certain optimality property of a one-sided test implicates that for a small level α the corresponding two-sided test acts only in a single direction. The results are applied to Kolmogorov–Smirnov type tests and the signal detection problem.