Power Robustification of Approximately Linear Tests

Abstract

We present a general method of improving the power of linear and approximately linear tests when deviations from a translation family of distributions must be taken into account. This method involves the combination of a linear statistic measuring location and a quadratic statistic measuring change of shape of the underlying distribution. The tests (“funnel tests”) are constructed as certain Bayes tests. In general they gain a sizeable amount of power over the linear tests adapted to the translation family when a change of shape of the underlying distribution occurs, while losing little for translation alternatives (“power robustification”). We introduce the concept of funnel tests in an Gaussian framework first. The effect of power robustification is studied 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 rotational cone. The idea of the construction is then carried over to a finite sample situation where the Gaussian model is used as an approximation. As a particular application, we construct power-robustified nonlinear rank tests in the standard two-sample situation. A simulation study demonstrates the good overall performance of these tests as compared to other nonlinear tests.

Key Words:

• Bayes test
• Conical alternatives
• Dimensional asymptotics
• Funnel test
• Hellinger geometry
• Location-shape alternatives
• Nonlinear rank test
• Nonparametric test
• Omnibus test
• Power comparison