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Inequalities are key tools to prove FDR control of a multiple test. Thepresent paper studies upper and lower bounds for the FDR under variousdependence structures of p-values, namely independence, reverse martingaledependence and positive regression dependence on the subset (PRDS) of true nullhypotheses. The inequalities are based on exact finite sample formulas whichare also of interest for independent uniformly distributed p-values under thenull. As applications the asymptotic worst case FDR of step up and step downtests coming from an non-decreasing rejection curve is established. Inaddition, new step up tests are established and necessary conditions for theFDR control are discussed. The reverse martingale models yield sharper FDRresults than the PRDS models. Already in certain multivariate normal dependencemodels the familywise error rate of the Benjamini Hochberg step up test can bedifferent from the desired level alpha. The second part of the paper is devotedto adaptive step up tests under dependence. The well-known Storey estimator ismodified so that the corresponding step up test has finite sample control forvarious block wise dependent p-values. These results may be applied todependent genome data. Within each chromosome the p-values may be reversemartingale dependent while the chromosomes are independent.

Inequalities for the false discovery rate (FDR) under dependence