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Weighted Asymptotically Optimal Sequential Testing

Soumyabrata Bose, Jay Bartroff

Abstract

This paper develops a framework for incorporating prior information into sequential multiple testing procedures while maintaining asymptotic optimality. We define a weighted log-likelihood ratio (WLLR) as an additive modification of the standard LLR and use it to construct two new sequential tests: the Weighted Gap and Weighted Gap-Intersection procedures. We prove that both procedures provide strong control of the family-wise error rate. Our main theoretical contribution is to show that these weighted procedures are asymptotically optimal; their expected stopping times achieve the theoretical lower bound as the error probabilities vanish. This first-order optimality is shown to be robust, holding in high-dimensional regimes where the number of null hypotheses grows and in settings with random weights, provided that mild, interpretable conditions on the weight distribution are met.

Weighted Asymptotically Optimal Sequential Testing

Abstract

This paper develops a framework for incorporating prior information into sequential multiple testing procedures while maintaining asymptotic optimality. We define a weighted log-likelihood ratio (WLLR) as an additive modification of the standard LLR and use it to construct two new sequential tests: the Weighted Gap and Weighted Gap-Intersection procedures. We prove that both procedures provide strong control of the family-wise error rate. Our main theoretical contribution is to show that these weighted procedures are asymptotically optimal; their expected stopping times achieve the theoretical lower bound as the error probabilities vanish. This first-order optimality is shown to be robust, holding in high-dimensional regimes where the number of null hypotheses grows and in settings with random weights, provided that mild, interpretable conditions on the weight distribution are met.

Paper Structure

This paper contains 17 sections, 8 theorems, 60 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Setup and Notations
  3. Background and Related Work
  4. Weighted Sequential Procedures
  5. Weighted Gap Procedure for Known Signal Count
  6. Weighted Gap-Intersection Procedure for Bounded Signal Count
  7. Asymptotic Optimality Analysis
  8. Asymptotic Optimality of the Weighted Gap Procedure
  9. Asymptotic Optimality of the Weighted Gap-Intersection Procedure
  10. Regime of Random Weights
  11. Simulation Study
  12. Simulation Setup
  13. Results
  14. Conclusion
  15. Appendix
  16. ...and 2 more sections

Key Result

Proposition 4.1

For any true signal set $A$ with $|A| = m$, the FWE of the weighted gap procedure $(T_W, D_W)$ is bounded above by Consequently, to ensure $\sup_{|A| = m} \mathbb{P}_A(D_W \neq A) \le \alpha$, it suffices to choose the threshold $c$ such that where the term $\mathcal{C}_W(m, J)$ is defined as

Figures (1)

  • Figure 1: Impact of Weight Informativeness on Expected Stopping Time

Theorems & Definitions (30)

  • Definition 2.1: Class of Admissible Procedures
  • Definition 4.1: Weighted Gap Procedure
  • Proposition 4.1: FWE Control for Weighted Gap Procedure
  • Remark 4.1: The Price of Weighting
  • Remark 4.2: Computation of $\mathcal{C}_W(m, J)$
  • Remark 4.3: Invariance to Scaling
  • Definition 4.2: Weighted Gap-Intersection Procedure
  • Proposition 4.2: FWE Control for Weighted Gap-Intersection Procedure
  • Remark 4.4: Behavior at Interval Boundaries
  • Remark 4.5: Interpretation of Thresholds
  • ...and 20 more