e-HC: Adaptive Sequential Higher Criticism Test for Sparse Mixtures

Gabriel Moryoussef

Keywords: higher criticism, sequential test, supermartingale, sparse mixture, Ville's inequality

Abstract: We propose e-HC, an adaptive sequential test for detecting sparse and weak signals in a stream of p-values. Unlike existing approaches that rely on asymptotic approximations or require knowledge of alternative parameters, e-HC constructs exact test-martingales using moment-generating function compensators, ensuring anytime-valid Type I error control through Ville's inequality. The method adapts to unknown sparsity and signal strength by maintaining exponential weights across multiple detection thresholds, effectively learning the optimal threshold online. We establish non-asymptotic power guarantees for sparse Gaussian mixtures alternative and derive the expected stopping time scaling for weak signal regimes. The same martingale machinery naturally yields anytime-valid confidence sequences for the proportion of significant p-values. Simulations demonstrate that e-HC maintains robust performance under model misspecification, substantially outperforming sequential likelihood ratio tests when the true alternative differs from assumptions.

Primary Area: other topics in machine learning (i.e., none of the above)

Submission Number: 12503