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Change, dependence, and discovery: Celebrating the work of T.L. Lai

Alexander G. Tartakovsky, Jay Bartroff, Cheng-Der Fuh, Haipeng Xing

TL;DR

This article surveys Tze Leung Lai's foundational work in sequential analysis, changepoint detection, and nonlinear renewal theory, with emphasis on non i.i.d. data and applications to biostatistics. It highlights near optimality results such as the SPRT being first-order asymptotically optimal as $\alpha_{\max}\to 0$ when the normalized LLR $n^{-1}\lambda_n$ converges to finite limits $I_i$, and the 2-SPRT under similar conditions. It also presents Lai's GLR tests for composite hypotheses, adaptive time varying boundaries, and extensions to multi-parameter exponential families, as well as Bayesian and uniform asymptotic optimality results. The review underscores practical impact on online learning, real time monitoring, changepoint detection, and adaptive clinical trial designs, illustrating a unifying framework that connects hypothesis testing, change detection, and sequential experimentation.

Abstract

Tze Leung Lai made seminal contributions to sequential analysis, particularly in sequential hypothesis testing, changepoint detection and nonlinear renewal theory. His work established fundamental optimality results for the sequential probability ratio test and its extensions, and provided a general framework for testing composite hypotheses. In changepoint detection, he introduced new optimality criteria and computationally efficient procedures that remain influential. He applied these and related tools to problems in biostatistics. In this article, we review these key results in the broader context of sequential analysis.

Change, dependence, and discovery: Celebrating the work of T.L. Lai

TL;DR

This article surveys Tze Leung Lai's foundational work in sequential analysis, changepoint detection, and nonlinear renewal theory, with emphasis on non i.i.d. data and applications to biostatistics. It highlights near optimality results such as the SPRT being first-order asymptotically optimal as when the normalized LLR converges to finite limits , and the 2-SPRT under similar conditions. It also presents Lai's GLR tests for composite hypotheses, adaptive time varying boundaries, and extensions to multi-parameter exponential families, as well as Bayesian and uniform asymptotic optimality results. The review underscores practical impact on online learning, real time monitoring, changepoint detection, and adaptive clinical trial designs, illustrating a unifying framework that connects hypothesis testing, change detection, and sequential experimentation.

Abstract

Tze Leung Lai made seminal contributions to sequential analysis, particularly in sequential hypothesis testing, changepoint detection and nonlinear renewal theory. His work established fundamental optimality results for the sequential probability ratio test and its extensions, and provided a general framework for testing composite hypotheses. In changepoint detection, he introduced new optimality criteria and computationally efficient procedures that remain influential. He applied these and related tools to problems in biostatistics. In this article, we review these key results in the broader context of sequential analysis.
Paper Structure (23 sections, 21 theorems, 165 equations, 1 figure)

This paper contains 23 sections, 21 theorems, 165 equations, 1 figure.

Table of Contents

  1. INTRODUCTION
  2. Biographical Sketch
  3. The Remainder of this Article
  4. SEQUENTIAL HYPOTHESIS TESTING
  5. Near Optimality of Wald's SPRT for Non-i.i.d. Models
  6. Near Optimality of Lorden's 2-SPRT for Non-i.i.d. Models
  7. Near Optimality of GLR Sequential Tests for Composite Hypotheses
  8. QUICKEST CHANGE DETECTION AND ISOLATION
  9. Lai's Changepoint Detection Theory for Non--i.i.d. Data
  10. Lai's Changepoint Detection--Isolation Theory for Non--i.i.d. Data
  11. MULTIPLE CHANGE-POINT SURVEILLANCE AND ESTIMATION
  12. An Extended Shiryaev's Rule for Composite Hypotheses
  13. Surveillance of Multiple Change-Points
  14. Inference on Multiple Change Points
  15. NONLINEAR RENEWAL THEORY
  16. ...and 8 more sections

Key Result

Theorem 1

Let $r > 0$. Assume that there exist finite constants $I_0 < 0$ and $I_1 > 0$ such that $n^{-1} \lambda_n$ converges $r$-quickly to $I_i$ under ${\mathsf{P}}_i$ for $i = 0, 1$, i.e., the conditions rquickLLR are satisfied. Then, as ${\alpha_{\rm max}} \to 0$,

Figures (1)

  • Figure 1: Left: Lai in his office at Columbia University, early 1970s. Right: Lai presenting a talk at the IMS-FIPS meeting, Shanghai, June 2019.

Theorems & Definitions (23)

  • Theorem 1: SPRT Asymptotic Optimality
  • Theorem 2: 2-SPRT Asymptotic Optimality
  • Remark 1
  • Theorem 3: Bayesian Optimality with Indifference Zone
  • Theorem 4: Bayesian Optimality, No Indifference Zone
  • Theorem 5: Uniform Asymptotic Optimality
  • Theorem 6: CUSUM Asymptotic Optimality, Non-i.i.d.
  • Theorem 7
  • Theorem 8
  • Remark 2
  • ...and 13 more