Large-Scale Two-Sample Comparison of Support Sets

ABSTRACT

Two-sample multiple testing has a wide range of applications. Most of the literature considers simultaneous tests of equality of parameters. The article takes a different perspective and investigates the null hypotheses that the two support sets are equal. This formulation of the testing problem is motivated by the fact that in many applications where the two parameter vectors being compared are both sparse, one might be more concerned about the detection of differential sparsity structures rather than the difference in parameter magnitudes. Focusing on this type of problem, we develop a general approach, which adapts the newly proposed symmetry data aggregation tool combined with a novel double thresholding (DT) filter. The DT filter first constructs a sequence of pairs of ranking statistics that fulfill global symmetry properties and then chooses two data-driven thresholds along the ranking to simultaneously control the False Discovery Rate (FDR) and maximize the number of rejections. Several applications of the methodology are given including high-dimensional linear models and Gaussian graphical models. We show that the proposed method is able to asymptotically control the FDR and have power guarantee under certain conditions. Numerical results confirm the effectiveness and robustness of DT in FDR control and detection ability. Supplementary materials for this article are available online.

KEYWORDS:

• Data-driven methods
• Gaussian graphical models
• High-dimensional models
• Multiple testing
• Sample-splitting
• Uniform convergence

Supplementary Materials

The Supplementary Material contains the proofs of Theorems 3.1, 3.2, Corollaries 4.1 and 4.2, and some additional numerical results.

Acknowledgments

The authors thank the Editor, Associate Editor and three anonymous referees for their many helpful comments that have resulted in significant improvements in the article.

Additional information

Funding

This research was supported by the China National Key R&D Program (Grant Nos. 2019YFC1908502, 2022YFA1003703, 2022YFA1003802, 2022YFA1003803), NNSF of China grants (nos. 11925106, 12231011, 11931001, 11971247, and 1210010142) and Shanghai Sailing Program.