Dynamic adaptive procedures for false discovery rate estimation and control
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In the multiple testing problem with independent tests, the classical linear step-up procedure controls the false discovery rate (FDR) at level π_0α, where π_0 is the proportion of true null hypotheses and α is the target FDR level. The adaptive procedures can improve power by incorporating estimates of π_0, which typically rely on a tuning parameter. For the fixed adaptive procedures that set their tuning parameters before seeing the data, the properties of the conservative FDR estimation and finite sample FDR control hold simultaneously. Less is known about the dynamic adaptive procedures whose tuning parameters are determined by the data. We show that, if the tuning parameter is chosen according to a left-to-right stopping time (LRS) rule from a fixed candidate set, the corresponding adaptive procedure estimates the FDR conservatively and controls the FDR as well. Furthermore, even if the tuning parameter candidates are random, such as p-values, we are able to prove the finite sample control of the FDR by using a series of LRS approximations. Consequently, the finite sample FDR control property is established for most existing dynamic adaptive procedures.
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