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Large-scale multiple testing is a fundamental problem in high dimensionalstatistical inference. It is increasingly common that various types ofauxiliary information, reflecting the structural relationship among thehypotheses, are available. Exploiting such auxiliary information can booststatistical power. To this end, we propose a framework based on a two-groupmixture model with varying probabilities of being null for different hypothesesa priori, where a shape-constrained relationship is imposed between theauxiliary information and the prior probabilities of being null. An optimalrejection rule is designed to maximize the expected number of true positiveswhen average false discovery rate is controlled. Focusing on the orderedstructure, we develop a robust EM algorithm to estimate the prior probabilitiesof being null and the distribution of $p$-values under the alternativehypothesis simultaneously. We show that the proposed method has better powerthan state-of-the-art competitors while controlling the false discovery rate,both empirically and theoretically. Extensive simulations demonstrate theadvantage of the proposed method. Datasets from genome-wide association studiesare used to illustrate the new methodology.

Optimal False Discovery Rate Control for Large Scale Multiple Testing with Auxiliary Information