Regions of alternatives with high and low power for goodness-of-fit tests

In the present paper we find finite dimensional spaces W of alternatives with high power fora given class of tests and non-parametric alternatives. On the orthogonal complement of W the power function is flat. These methods can be used to reduce the dimension of interesting alternatives. We sketch a device how to calculate (approximately) an alternative with maximum power of a fixed test on a given ball of certain non-parametric alternatives. The calculations are done within different asymptotic models specified by signal detection tests. Specific tests are Kolmogorov-Smirnov type tests, integral tests (like the Anderson and Darling test) and Renyi tests for hazard based models. The statistical meaning and interpretation of the spaces of alternatives with high power is discussed. These alternatives belong to least favorable directions of a class of statistical functionals which are linear combinations of quantile functions. For various cases their meaning is explained for parametric submodels, in particular for location alternatives. (C) 2007 Elsevier B.V. All rights reserved.