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 som e generalizations of the test of Mack and Wolfe (1981). All the tests are b ased on pairwise ranking methods. The asymptotic power and asymptotic relat ive efficiency of the so-called Mack-Wolfe-type test, Tryon-Hettmannsperger -type-test and Purl-type test are derived and compared with modifications o f the test of Hettmansperger and Norton (1987) where the ranks are taken ov er 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 achiev able efficacies in the defined classes of the generalized Mack-Wolfe-type t est and Tryon-Hettmansperger-type test always are equal. The case of an unknown peak is briefly discussed.