Tchebycheff-Experiments

摘要

When testing n #x2014; 1 hypotheses versus a simple alternative the corresponding generalized Neyman Pearson (GNP) tests provide an optimal solution from a theoretical point of view. However, the practical merit of these tests depends heavily on its simplicity. When the sample space is completely ordered the most simplest tests are monotone, i.e., roughly speaking, the critical region consists in the union of at most n/2 intervals (here x denotes the smallest integer #x2264; x). We show that the existence of monotone GNP-tests in a dominated family of distributions implies that each selection of n densities constitute a weak Tchebycheff-system of order n. These experiments are denoted as weak Tchebycheff-experiments. In particular, we show that under mild topological assumptions on the parameter space weak Tchebycheff-experiments are sign regular, provided continuous versions of the densities exist. Further we determine the topological structure of the sample and the parameter space of Tchebycheff-experiments. Various examples of sign regular experiments on the real line and the circle are discussed. Finally, applications to distributions of directional data and complete class theorems are given. In particular, we indicate how this concept can be applied successfully to describe the shape of GNP-tests although the densities are not sign regular.