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 concerning the comparison of two popular multiple test procedures, so-called step-down and step-up procedures, in terms of their defining critical values. Such procedures can be applied, for example, to multiple comparisons with a control. In the definition of the critical values for these procedures order statistics play a central role. For $k \epsilon N_0$ fixed we consider the joint cumulative distribution function (cdf) $P(Y_{1:n} \leq c_1,\dots, Y_{n-k:n} \leq c_{n-k})$ of the first $n - k$ order statistics and the cdf$P(Y_{n-k:n} \leq c_{n-k})$ of the ($k + 1$)th largest order statistic $Y_{n-k:n}$ of $n$ random variables $Y_1,\dots, Y_n$ belonging to a sequence of exchangeable real-valued random variables. Our interest is focused on the asymptotic behavior of these cdf's and their interrelation if $n$ tends to $\infty$. It turns out that they sometimes behave completely differently compared with the iid case treated in Finner and Roters so that positive results are only possible under additional assumptions concerning the underlying distribution. We consider different sets of assumptions which then allow analogous results for the exchangeable case. Recently, Dalal and 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 iid case. We give a counterexample for the exchangeable case underlining the difficulties occurring in this situation. As an application we consider the comparison of certain step-down and step-up procedures in multiple comparisons with a control. The results of this paper yield a more theoretical explanation of the superiority of the step-up procedure which has been observed earlier by comparing tables of critical values. As a by-product we are able to quantify the tightness of the Bonferroni inequality in connection with maximum statistics.