With high dimensionality, standard Bonferroni-style procedures can suffer from loss of power, since the significance level $\alpha$ must be divided by $k$ to declare significance. Kropf and Läuter (KL) show that certain data-dependent quadratic forms can be used to "pre-specify hypotheses, which can then be tested in a fixed, data-dependent order, without multiplicity adjustment. In this article we extend the KL procedure to a class of weighted procedures, using the same quadratic forms. The class includes the KL method, the Bonferroni-Holm method, and other, new procedures. We establish strong FWE control for all procedures, and compare power and level of various weighting methods using analytical and simulation results. The method is applied using a high-dimensional mixture model that is suggested by the analysis of real gene expression data.
Digital Object Identifier: 10.1214/lnms/1196285632