Power robustification of approximately linear tests.

*(English)*Zbl 0843.62053Summary: 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 a 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.

We introduce the concept of funnel tests in a 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.

##### MSC:

62G10 | Nonparametric hypothesis testing |

62F15 | Bayesian inference |

62F03 | Parametric hypothesis testing |