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Abstract:
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Multiple testing is a fundamental problem in high-dimensional statistical inference. Although many false discovery control methods have been proposed, it is still a challenging task when the tests are correlated to each other. Fan et al. (2012) is a milestone paper for the estimation of the false discovery proportion (FDP) under arbitrary covariance dependence. Their method, FPA, reduces arbitrary covariance dependence to weak dependence by removing the dominant factors from spectral decomposition of the covariance matrix and then establishes a strongly consistent estimator of the FDP. In particular, the estimation of the FDP under weak dependence is identical to that under independence. In this article, we further study the asymptotic variance of the FDP under the same framework. We find that even under weak dependence, the asymptotic variance of the FDP can still differ dramatically from that under independence and the difference depends on the dependence structure between the tests. This result can potentially provide a confidence level for controlling the FDP, even when its mean, false discovery rate (FDR), is controlled at a given level.
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