POWER ROBUSTIFICATION OF APPROXIMATELY LINEAR TESTS

We present a general method of improving the power of linear and appro ximately linear tests when deviations from a translation family of dis tributions must be taken into account. This method involves the combin ation of a linear statistic measuring location and a quadratic statist ic measuring change of shape of the underlying distribution. The tests ( ''funnel tests'') are constructed as certain Bayes tests. In genera l they gain a sizeable amount of power over the linear tests adapted t o the translation family when a change of shape of the underlying dist ribution occurs, while losing little for translation alternatives (''p ower robustification''). We introduce the concept of funnel tests in a n Gaussian framework first. The effect of power robustification is stu died by means of a power function expansion, which applies to a large class of tests sharing a certain invariance property. The funnel tests are characterized by a maximin property over a region defined by a ro tational cone. The idea of the construction is then carried over to a finite sample situation where the Gaussian model is used as an approxi mation. As a particular application, we construct power-robustified no nlinear rank tests in the standard two-sample situation. A simulation study demonstrates the good overall performance of these tests as comp ared to other nonlinear tests.