On the evaluation of the joint distribution of order statistics

Dunnett and Tamhane [Dunnett, C.W., Tamhane, A.C., 1992. A step-up multiple test procedure. J. Amer. Statist. Assoc. 87, 162–170.] proposed a step-up procedure for comparing kk treatments with a control and showed that the step-up procedure is more powerful than its counterpart single step and step-down procedures. Since then, several modified step-up procedures have been suggested to deal with different testing environments. In order to establish those step-up procedures, it is necessary to derive approaches for evaluating the joint distribution of the order statistics. In some cases, experimenters may have difficulty in applying those step-up procedures in multiple hypothesis testing because of the computational limitation of existing algorithms in evaluating the critical values for a large number of multiple comparisons. As a result, most procedures are only workable when the design of the experiment is balanced with k≤20k≤20 or unbalanced with k≤8k≤8. In this paper, new algorithms are proposed in order to effectively compute the joint distribution of order statistics in various situations. An extensive numerical study shows that the proposed algorithms can easily handle the testing situations with a much larger kk. Examples of applying the proposed algorithms to evaluate the critical values of two existing step-up procedures are also presented.