Asymptotic comparison of step-down and step-up multiple test procedures based on exchangeable test statistics

In this paper interest is focused on some theoretical investigations concer ning the comparison of two popular multiple test procedures, so-called step -down and step-up procedures, in terms of their defining critical values. S uch procedures can be applied, for example, to multiple comparisons with a control. In the definition of the critical Values for these procedures orde r statistics play a central role. For k epsilon N-0 fixed we consider the j oint cumulative distribution function (cdf) P(Y-1:n less than or equal to c (1),...,Yn-k:n less than or equal to c(n-k)) of the first n - k order stati stics and the cdf P(Yn-k:n less than or equal to c(n-k)) of the (k + 1)th l argest order statistic Yn-k:n of n random variables Y-1,...,Y-n belonging t o a sequence of exchangeable real-valued random variables. Our interest is focused on the asymptotic behavior of these cdfs and their interrelation if n tends to infinity. It turns out that they sometimes behave completely di fferently compared with the lid case treated in Finner and Roters so that p ositive results are only possible under additional assumptions concerning t he underlying distribution. We consider different sets of assumptions which then allow analogous results for the exchangeable case. Recently, Dalal an d Mallows derived a result concerning the monotonicity of a certain set of critical values in connection with the joint cdf of order statistics in the lid case. We give a counterexample for the exchangeable case underlining t he difficulties occurring in this situation. As an application we consider the comparison of certain step-down and step-up procedures in multiple comp arisons with a control. The results of this paper yield a more theoretical explanation of the superiority of the step-up procedure which has been obse rved earlier by comparing tables of critical values. As a byproduct we are able to quantify the tightness of the Bonferroni inequality in connection w ith maximum statistics.