Large-Scale Two-Sample Comparison of Support Sets

Haoyu Geng, Xiaolong Cui, Haojie Ren, Changliang Zou

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.