A note on the monotonicity of the critical values of a step-up test

Abstract

Abstract The monotonicity of the critical values is essential for the implementation of a step-up multiple test procedure. For a step-up test controlling the familywise error rate exactly at α, while a theoretical justification of the existence of monotone critical values is available in the literature for the case of i.i.d. test statistics, it is still an open problem for the general exchangeable case. Recently, Finner and Roters (1998 Ann. Statist. 26 505–524) gave an example of three exchangeable random variables casting doubt about this existence in general. We provide a positive result in this paper by proving that the critical values of a step-up test involving an equicorrelated trivariate standard normal distribution is indeed monotone when the common correlation is positive and at most (zα/22−z1/42)/(zα/22+z1/42), where zα is the upper 100α percent point of N(0,1). Our example raises the hope of establishing the desired monotonicity property in the case of test statistics with totally positive multivariate distributions in the sense of Karlin and Rinott.