This paper extends the theory of false discovery rates (FDR) pioneered by Benjamini and Hochberg J. Roy. Statist. Soc. Ser. B 57 (1995) 289-300. We develop a framework in which the False Discovery Proportion (FDP)-the number of false rejections divided by the number of rejections-is treated as a stochastic process. After obtaining the limiting distribution of the process, we demonstrate the validity of a class of procedures for controlling the False Discovery Rate (the expected FDP). We construct a confidence envelope for the whole FDP process. From these envelopes we derive confidence thresholds, for controlling the quantiles of the distribution of the FDP as well as controlling the number of false discoveries. We also investigate methods for estimating the p-value distribution.
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