Sharp multiple testing boundary for sparse sequences
Kweku Abraham, Ismaël Castillo, Étienne Roquain
The Annals of Statistics · 2024 · Volume 52 · Issue 4
Abstract
This work investigates multiple testing by considering minimax separation rates in the sparse sequence model, when the testing risk is measured as the sum FDR+FNR (False Discovery Rate plus False Negative Rate). First, using the popular beta-min separation condition, with all nonzero signals separated from 0 by at least some amount, we determine the sharp minimax testing risk asymptotically and thereby explicitly describe the transition from “achievable multiple testing with vanishing risk” to “impossible multiple testing.” Adaptive multiple testing procedures achieving the corresponding optimal boundary are provided: the Benjamini–Hochberg procedure with a properly tuned level, and an empirical Bayes ℓ-value (‘local FDR’) procedure. We prove that the FDR and FNR make nonsymmetric contributions to the testing risk for most optimal procedures, the FNR part being dominant at the boundary. The multiple testing hardness is then investigated for classes of arbitrary sparse signals. A number of extensions, including results for classification losses and convergence rates in the case of large signals, are also investigated.
Keywords
62G15, 62G20, Benjamini–Hochberg procedure, False discovery rate, Frequentist analysis of Bayesian procedures, multiple testing, sharp asymptotic minimaxity
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