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In modern scientific experiments, we frequently encounter data that havelarge dimensions, and in some experiments, such high dimensional data arrivesequentially rather than full data being available all at a time. We developmultiple testing procedures with simultaneous control of false discovery andnondiscovery rates when $m$-variate data vectors $\mathbf{X}_1, \mathbf{X}_2,\dots$ are observed sequentially or in groups and each coordinate of thesevectors leads to a hypothesis testing. Existing multiple testing methods forsequential data uses fixed stopping boundaries that do not depend on samplesize, and hence, are quite conservative when the number of hypotheses $m$ islarge. We propose sequential tests based on adaptive stopping boundaries thatensure shrinkage of the continue sampling region as the sample size increases.Under minimal assumptions on the data sequence, we first develop a test basedon an oracle test statistic such that both false discovery rate (FDR) and falsenondiscovery rate (FNR) are nearly equal to some prefixed levels with strongcontrol. Under a two-group mixture model assumption, we propose a data-drivenstopping and decision rule based on local false discovery rate statistic thatmimics the oracle rule and guarantees simultaneous control of FDR and FNRasymptotically as $m$ tends to infinity. Both the oracle and the data-drivenstopping times are shown to be finite (i.e., proper) with probability 1 for allfinite $m$ and converge to a finite constant as $m$ grows to infinity. Further,we compare the data-driven test with the existing gap rule proposed in He andBartroff (2021) and show that the ratio of the expected sample sizes of ourmethod and the gap rule tends to zero as $m$ goes to infinity. Extensiveanalysis of simulated datasets as well as some real datasets illustrate thesuperiority of the proposed tests over some existing methods.

Large-scale adaptive multiple testing for sequential data controlling false discovery and nondiscovery rates