SCORE: A Unified Framework for Overshoot Refund in Online FDR Control

TL;DR

The paper addresses robust online FDR control with $e$-values by identifying and exploiting overshoot from strong rejections. It introduces the SCORE framework, which refunds overshoot via the overshoot term $O_j=(\alpha_j e_j-1)_+$, yielding SCORE-LOND, SCORE-LORD, and SCORE-SAFFRON that strictly dominate their predecessors while preserving finite-sample FDR control. Under CPQD, SCORE enables retroactive wealth updates (SCORE$^+$ variants) by using a global denominator $R_t\vee 1$, further boosting power. The framework is validated through extensive simulations and a real yeast dataset, demonstrating consistent improvements in power with maintained FDR control. Overall, SCORE provides a universal, theoretically sound enhancement to online hypothesis testing that can be integrated with existing online testing algorithms and extended to broader dependence settings.

Abstract

We propose a unified framework to enhance the power of online multiple hypothesis testing procedures based on $e$-values. While $e$-value-based methods offer robust online False Discovery Rate (FDR) control under minimal assumptions, they often suffer from power loss by discarding evidence that exceeds the rejection threshold. We address this inefficiency via the \textbf{S}equential \textbf{C}ontrol with \textbf{O}vershoot \textbf{R}efund for \textbf{E}-values (SCORE) framework, which leverages the inequality $\mathbb{I}(y \ge 1) \le y - (y-1)_+$ to reclaim this otherwise ``wasted'' evidence. This simple yet powerful insight yields a unified principle for improving a broad class of online testing algorithms. Building on this framework, we develop SCORE-enhanced versions of several state-of-the-art procedures, including SCORE-LOND, SCORE-LORD, and SCORE-SAFFRON, all of which strictly dominate their original counterparts while preserving valid finite-sample FDR control. Furthermore, under mild assumptions, SCORE permits retroactive updates of alpha-wealth by using the latest decision twice: first to determine its reward or loss, and then to refresh past wealth. Such a mechanism enables more aggressive testing strategies while maintaining valid FDR control, thereby further improving statistical power. The effectiveness of the proposed methods is validated through extensive simulation and real-data experiments.

Figures (6)

• Figure 1: Independent setting: SCORE methods increase power while maintaining FDR control across varying non-null proportions.
• Figure 2: Dependent setting. SCORE and SCORE$^+$ variants exhibit higher power than the base methods.
• Figure 3: Power and FDR comparison of 12 methods under the AR(1) setting: 6 $e$-value-based methods and 6 classical $p$-value methods using conditional $p$-values. The dashed red line represents the target FDR level $\alpha=0.05$. All methods successfully control FDR.
• Figure 4: Evolution of Power and FDR over time for 12 methods under the AR(1) setting: 6 $e$-value-based methods and 6 classical $p$-value methods using $p$-values. The dashed red line represents the target FDR level $\alpha=0.05$.
• Figure 5: Power and FDR comparison of the same 6 classical $p$-value methods when using marginal $p$-values under the AR(1) setting. The dashed red line represents the target FDR level $\alpha=0.05$. Note that all methods fail to control FDR, demonstrating that marginal validity is insufficient under temporal dependence.

Theorems & Definitions (30)

• Lemma 1
• Theorem 1
• Proposition 1
• Proposition 2
• Proposition 3
• Proposition 4
• Theorem 2
• Proposition 5
• Theorem 3
• Proposition 6