Multiple Comparison Procedures in Multi-Sample Exponential Models

We construct multiple comparisons procedures in k exponential populations. Exact theory and asymptotic theory of simultaneous confidence intervals and multiple comparisons tests are discussed. First, we consider multiple comparisons for the differences among parameters. We can give the Tukey-Kramer type multiple test procedure based on estimators of k means. However, the degree of conservativeness for the multiple tests depends on unknown mean parameters. Therefore, multiple tests based on the logarithm transformation of estimators are proposed. It is found that the degree of conservativeness for the proposed tests is controlled by the sample sizes. Furthermore, the closed testing procedure, more powerful than the REGW (Ryan/Einot-Gabriel/Welsch) tests, is proposed. Simultaneous confidence intervals for the differences among the logarithms of parameters are discussed. Next, for the multiple comparisons with a control, we propose the multiple test procedures. It is shown that the proposed multiple test is superior to the tests based on the Bonferroni inequality asymptotically. A sequentially rejective procedure is derived under unequal sample sizes. Last, we consider multiple comparisons for all parameters. The exact single-step multiple comparison procedures based on the upper 100α% points the χ2-distribution are proposed. The asymptotic theory for the multiple comparisons is discussed. Especially sequentially rejective procedures can be constructed in the asymptotic theory.