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The Asymptotic Power and Relative Efficiency of Some c-Sample Rank Tests of Homogeneity against Umbrella Alternatives

For the c-sample location problem with umbrella alternatives we compare some generalizations of the test of Mack and Wolfe (1981). All the tests are based on pairwise ranking methods. The asymptotic power and asymptotic relative efficiency of the so-called Mack-Wolfe-type test, Tryon-Hettmannsperger-type-test and Puri-type test are derived and compared with modifications of the test of Hettmansperger and Norton (1987) where the ranks are taken over all samples. The Mack-Wolfe-type test and the Tryon-Hettmansperger-type-test are further generalized by introducing weight coefficients for the substatistics. For the case of a specified alternative these weights are determined in such a way that the efficacies become maximal. It is shown that the maximal achievable efficacies in the defined classes of the generalized Mack-Wolfe-type test and Tryon-Hettmansperger-type test always are equal. The case of an unknown peak is briefly discussed.