Modifications of the Weighted Mack-Wolfe Umbrella Tests for a Generalized Behrens-Fisher Problem

Modifications of t he weighted Mack-Wolfe tests are proposed for both cases when the peak of umbrella is known and unknown. The modified weighted Mack-Wolfe tests are exactly distribution-free when the continuous populations have the same shape. For the case of peak-known umbrella alternatives, the modified weighted Mack-Wolfe tests remain asymptotically distribution-free when the continuous populations are symmetric, but not necessarily with the same shape. For the case of peak-unknown umbrella alternatives, the maximum of standardized modified weighted Mack-Wolfe tests with peak-known umbrella alternatives was used. The simulation results show that the modified weighted Mack-Wolfe tests are more recommended than the modified Mack-Wolfe tests for various patterns.