Some c-sample rank tests of homogeneity against umbrella alternatives with unknown peak

Abstract

The c-sample location problem with umbrella alternatives is investigated. The umbrella peak l is assumed to be unknown. All test statistics are maxima of substatistics which are constructed for the case of a known peak l. These substatistics are based on ranks. Arbitrary scores and arbitrary sample size configurations are allowed. As special cases we have tests of Chen and Wolfe [Chen, Y.I. and Wolfe, D.A., 1990, A study of distribution-free tests for umbrella alternatives. Biometrical Journal, 32, 47–57.], Hettmansperger and Norton [Hettmansperger, T.P. and Norton, R.M., 1987, Tests for patterned alternatives in k-sample problems. Journal of the American Statistical Association, 82, 292–299.] and Shi [Shi, N.-Z., 1988, Rank test statistics for umbrella alternatives. Communications in Statistics, Theory and Methods, 17, 2059–2073.]. We compute the asymptotic power functions of the constructed tests. For the cases of three or four treatments, some figures give an impression of the asymptotic power functions for various setups. Simulation studies show that the asymptotic results can also be used for moderate sample sizes. In average, over all setups, the Hettmansperger–Norton-type test performs best densely followed by the Chen–Wolfe-type test and the Shi-type test.