A New Non-Parametric Test for Unbrella Alternatives
The Mann-Whitney statistics have commonly been used as a building block for many tests involving ordered and umbrella alternatives. Such tests are based on pair wise information that is obtained from all Ck 2 pairs of samples. This dissertation introduces a new nonparametric test for testing umbrella alternatives in a completely randomized one way design. This test is based on information obtained from a subset of the Ck 3 trios of samples. Unlike most existing tests for umbrella alternatives, the new test lays emphasis on the importance of testing across the peak of the umbrella, thereby rendering the new test more efficient. The test has the flexibility of testing other patterned alternatives such as the monotone ordering of location parameters (increasing or decreasing). The mean and variance of the test statistic are derived under the null hypothesis with the extensive details of the derivation included in the write-up. I also present a simplified mean and variance result that is in a practical form. Based on the derivation of the asymptotic distribution, the standardized test statistic corresponding to the new test converges in distribution to a standard normal distribution. Some numerical examples involving clinical data are analyzed. A simulation study compared power estimates of the new tests under different sample sizes and location parameters to those of seven other existing tests. The new test generally competed with all other existing tests and performed better than all the other tests in many scenarios.