Modifications of the Mack‐Wolfe umbrella tests for a generalized Behrens‐Fisher problem

We are concerned with testing procedures for umbrella alternatives in the k‐sample location problem without making the assumption that the underlying populations have the same shape. Modifications of the Mack‐Wolfe tests are proposed for the cases when the peak of the umbrella is known or unknown. The proposed procedures are exactly distribution‐free when the continuous populations have the same shape. The modified test for peak‐known umbrella alternatives remains asymptotically distribution‐free when the continuous populations are symmetric, but not necessarily with the same shape.