Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Asymptotically Optimal Sequential Multiple Testing Procedures for Correlated Normal

Monitirtha Dey, Subir Kumar Bhandari

TL;DR

This work investigates sequential multiple testing for $K$ equicorrelated Gaussian streams, where each stream tests $H_{0i}: X_{ij} o N(0,1)$ versus $H_{1i}: X_{ij} o N(\mu_i,1)$ with common correlation $ ho$. It develops gap-based sequential procedures that connect to SPRT and establish asymptotic optimality for controlling $FWER$ and a broad class of multiple-testing error metrics, including $FDR$ and $pFDR$, across known and bounded numbers of signals. The authors extend to unknown signal strength and correlation via consistent estimators, showing the procedures retain ASN optimality; simulations confirm that higher correlation reduces the average sample size and that the theory matches empirical performance as error levels vanish. These results provide principled, scalable strategies for dependent, high-dimensional sequential testing with practical implications for streaming inference and multi-sample decision making.

Abstract

Simultaneous statistical inference has been a cornerstone in the statistics methodology literature because of its fundamental theory and paramount applications. The mainstream multiple testing literature has traditionally considered two frameworks: the sample size is deterministic, and the test statistics corresponding to different tests are independent. However, in many modern scientific avenues, these assumptions are often violated. There is little study that explores the multiple testing problem in a sequential framework where the test statistics corresponding to the various streams are dependent. This work fills this gap in a unified way by considering the classical means-testing problem in an equicorrelated Gaussian and sequential framework. We focus on sequential test procedures that control the type I and type II familywise error probabilities at pre-specified levels. We establish that our proposed test procedures achieve the optimal expected sample sizes under every possible signal configuration asymptotically, as the two error probabilities vanish at arbitrary rates. Towards this, we elucidate that the ratio of the expected sample size of our proposed rule and that of the classical SPRT goes to one asymptotically, thus illustrating their connection. Generalizing this, we show that our proposed procedures, with appropriately adjusted critical values, are asymptotically optimal for controlling any multiple testing error metric lying between multiples of FWER in a certain sense. This class of metrics includes FDR/FNR, pFDR/pFNR, the per-comparison and per-family error rates, and the false positive rate.

Asymptotically Optimal Sequential Multiple Testing Procedures for Correlated Normal

TL;DR

This work investigates sequential multiple testing for equicorrelated Gaussian streams, where each stream tests versus with common correlation . It develops gap-based sequential procedures that connect to SPRT and establish asymptotic optimality for controlling and a broad class of multiple-testing error metrics, including and , across known and bounded numbers of signals. The authors extend to unknown signal strength and correlation via consistent estimators, showing the procedures retain ASN optimality; simulations confirm that higher correlation reduces the average sample size and that the theory matches empirical performance as error levels vanish. These results provide principled, scalable strategies for dependent, high-dimensional sequential testing with practical implications for streaming inference and multi-sample decision making.

Abstract

Simultaneous statistical inference has been a cornerstone in the statistics methodology literature because of its fundamental theory and paramount applications. The mainstream multiple testing literature has traditionally considered two frameworks: the sample size is deterministic, and the test statistics corresponding to different tests are independent. However, in many modern scientific avenues, these assumptions are often violated. There is little study that explores the multiple testing problem in a sequential framework where the test statistics corresponding to the various streams are dependent. This work fills this gap in a unified way by considering the classical means-testing problem in an equicorrelated Gaussian and sequential framework. We focus on sequential test procedures that control the type I and type II familywise error probabilities at pre-specified levels. We establish that our proposed test procedures achieve the optimal expected sample sizes under every possible signal configuration asymptotically, as the two error probabilities vanish at arbitrary rates. Towards this, we elucidate that the ratio of the expected sample size of our proposed rule and that of the classical SPRT goes to one asymptotically, thus illustrating their connection. Generalizing this, we show that our proposed procedures, with appropriately adjusted critical values, are asymptotically optimal for controlling any multiple testing error metric lying between multiples of FWER in a certain sense. This class of metrics includes FDR/FNR, pFDR/pFNR, the per-comparison and per-family error rates, and the false positive rate.
Paper Structure (13 sections, 7 theorems, 71 equations, 2 figures, 1 table)

This paper contains 13 sections, 7 theorems, 71 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The Testing Framework
  4. Asymptotic Optimality for controlling FWER
  5. Asymptotic Expansion of Expected Sample Size of One-sided SPRT
  6. Main Results for FWER Controlling Tests
  7. A distributionally equivalent representation of the vector observations
  8. Proposed procedure for known number of signals
  9. Proposed procedure when upper and lower bounds on the number of signals are available
  10. General Error Rate controlling Procedures
  11. The case with unknown mean and equicorrelation
  12. Simulation Study
  13. Concluding Remarks

Key Result

theorem 1

(Siegmund, page 20) Consider the class of all fixed-sample-size or sequential tests for the problem HTP 2, having the type I error probability not more than $\alpha<1$. In other words, the one-sided SPRT with type I error probability $\alpha$ minimizes $\mathbb{E}_{H_1}(T)$ in this class when $\alpha \to 0$.

Figures (2)

  • Figure 1: Estimates of average sample numbers of our proposed gap rule for different combinations of $(\rho, \alpha)$. Based on 1000 repetitions.
  • Figure 2: Ratios of the ASNs and the theoretical approximations of our proposed gap rule for different combinations of $(\rho, \alpha)$. Based on 1000 repetitions.

Theorems & Definitions (11)

  • remark 1
  • definition 1
  • theorem 1
  • theorem 2
  • theorem 3
  • theorem 4
  • theorem 5
  • proof
  • remark 2
  • theorem 6
  • ...and 1 more