  Large-scale multiple testing is a fundamental problem in high dimensional
statistical inference. It is increasingly common that various types of
auxiliary information, reflecting the structural relationship among the
hypotheses, are available. Exploiting such auxiliary information can boost
statistical power. To this end, we propose a framework based on a two-group
mixture model with varying probabilities of being null for different hypotheses
a priori, where a shape-constrained relationship is imposed between the
auxiliary information and the prior probabilities of being null. An optimal
rejection rule is designed to maximize the expected number of true positives
when average false discovery rate is controlled. Focusing on the ordered
structure, we develop a robust EM algorithm to estimate the prior probabilities
of being null and the distribution of $p$-values under the alternative
hypothesis simultaneously. We show that the proposed method has better power
than state-of-the-art competitors while controlling the false discovery rate,
both empirically and theoretically. Extensive simulations demonstrate the
advantage of the proposed method. Datasets from genome-wide association studies
are used to illustrate the new methodology.
