Large-scale Asymptotics in Fixed-sample and Sequential Multiple Testing

Abstract

The advent of large-scale data acquisition technologies has led to the routine emergence of massive, dynamically evolving datasets. This has created a pressing need for statistical methodologies capable of operating effectively in both static and sequential data environments. Multiple hypotheses testing, a central tool in such settings, must therefore be adapted to both fixed-sample and sequential frameworks to ensure effective decision-making while controlling error rates. This thesis comprises two main parts. The first part addresses fixed-sample multiple testing under dependence in a Bayesian framework. Building upon the two-group mixture normal model of Bogdan et al. (2011), we extend their methodology to exchangeable multivariate normal test statistics, thereby accommodating realistic dependency structures frequently encountered in high-dimensional applications. We derive sufficient conditions under which multiple testing procedures satisfy the Asymptotic Bayes Optimality under Sparsity (ABOS) property in the presence of dependence. Furthermore, we show that several classical procedures, including those of Bonferroni (1936), Šidák (1967), and Benjamini and Hochberg (1995), retain the ABOS property under suitable sparsity and dependence assumptions. The second part focuses on large-scale multiple testing in sequential settings, where data vectors or multiple data streams are observed over time rather than being available in full initially. Motivated by diverse applications, we investigate two distinct sampling termination schemes: (i) synchronous termination, in which sampling and testing for all hypotheses stop simultaneously; and (ii) asynchronous termination, in which different hypotheses may stop at different times. For the synchronous case, we develop the Oracle Intersection (OI) test, based on the local false discovery rate statistic under a two-group mixture model. The OI test guarantees exact control of both the False Discovery Rate (FDR) and the False Non-Discovery Rate (FNR) at prespecified levels. We further propose a fully datadriven version that achieves asymptotic simultaneous control of FDR and FNR as the number of hypotheses m → ∞. While existing sequential tests use fixed stopping boundaries, the proposed tests employ stopping boundaries that adapt quickly with the samplesize, ensuring shrinkage of the continue-sampling region as the trial progresses. As a result, stopping times for both the oracle and data-driven tests converge to a finite constant as m → ∞. Moreover, the ratio of the expected sample size of the OI test to that of the Gap rule (He and Bartroff, 2021) converges to zero as m →∞. Extensive analyses of real and simulated datasets demonstrate the superiority of the proposed methods over existing approaches. Finally, we address the case of asynchronous termination and introduce the Oracle Stagewise (OS) test, constructed from the local false discovery rate statistic under a two-group mixture model. Employing adaptive stopping boundaries, the OS test drops hypotheses by accepting or rejecting them at interim stages of sampling until all hypotheses are decided. This stagewise procedure achieves simultaneous control of the FDR and FNR, while substantially reducing the total sample size relative to existing sequential methods (Bartroff and Song, 2020). To address challenges arising from composite hypotheses and the instability of parameter estimation caused by shrinking active sets, we further propose the Oracle Composite (OC) and Data-driven Composite (DC) tests. These hybrid procedures combine stagewise elimination with intersection-based testing, ensuring reliable estimation and improved practical applicability. Simulation studies demonstrate that the proposed methods substantially reduce the total sample size relative to existing sequential procedures while maintaining rigorous error control. Overall, this thesis advances the theory of large-scale multiple testing in both fixed-sample and sequential settings by establishing new optimality results under dependence and developing adaptive procedures that simultaneously achieve rigorous error control, finite stopping times, and improved sampling efficiency.