Robust Umbrella Tests for a Generalized Behrens‐Fisher Problem

This paper is concerned with testing for umbrella alternatives in a k‐sample location problem when the underlying populations have possibly different shapes. Following CHEN and WOLFE (1990b), rank‐based modifications of the HETTMANSPERGER‐NORTON (1987) tests are considered for both the settings where the peak of the umbrella is known and where it is unknown. The proposed procedures are exactly distribution‐free when the continuous populations are identical with any shape. Moreover, the modified test for peak‐known umbrella alternatives remains asymptotically distribution‐free when the continuous populations are assumed to be symmetric, even if they differ in shapes. Comparative results of a Monte Carlo study are presented.