On the critical value behavior of multiple decision procedures (Q2742767)

The authors investigate the critical value behaviour of several multiple decision procedures when the number \(n\) of parameters or hypotheses under consideration increases. The authors obtain results of the following type. Suppose that the sequences of critical values to be compared are denoted by \(v_n\) and \(w_n.\) Since it is almost impossible to give explicit formulas for the critical values, and much less for the differences of these, the authors first derive asymptotic expansions for the critical points by means of a special method. In the majority of cases, these asymptotics do not provide enough information to yield an appropriate asymptotic expansion for the differences of the \(w_n\) and \(v_n\) by simple substraction. The authors employ the knowledge of the asymptotic behaviour of the corresponding differences based on uniformly distributed observations such as, for example, \(p\)-values to obtain a sequence \(g_n\) such that \(\lim_{n \to \infty} g_n(w_n - v_n) =g\) for some positive constant \(g.\)