Covariance-Adaptive Residualization and Stagewise Calibration for Dependent Multiple Testing

Abstract

In this paper, we study simultaneous hypothesis testing for multivariate Gaussian means under arbitrary covariance dependence. Building upon the Maximum Residual Down (MRD) procedure of Cohen et al. (2009), we investigate a systematic stagewise calibration strategy based on the generalized step-down critical constants of Gavrilov et al. (2009). The proposed methodology retains the covariance-adaptive residualization mechanism of MRD while replacing the original model-dependent threshold specification with a simple and principled calibration rule. Since the resulting procedure belongs to the class of monotone residual-based step-down procedures studied by Ghosh and Chakrabarti (2026), its admissibility follows directly from their general theory. We also derive alternative representations of the MRD residual statistics that express all active residuals through a single active precision matrix, substantially reducing computational complexity while revealing a direct connection between covariance-adaptive residualization and active precision-matrix geometry. Extensive simulation studies under a broad range of dependence structures demonstrate that the proposed methodology frequently achieves substantially lower normalized misclassification risk than several widely used marginal testing procedures. Under several structured dependence models, it also exhibits remarkably strong signal-recovery behavior, simultaneously attaining false discovery rates close to the nominal level, extremely small false non-discovery rates, powers approaching one, and average numbers of rejections close to the expected number of true signals. These findings suggest that covariance-adaptive residualization and stagewise calibration play complementary roles in exploiting dependence information for large-scale multiple testing under arbitrary covariance structures.