On the optimality of coin-betting for mean estimation
Proves coin-betting as the optimal e-variable strategy for sequential mean testing, establishing its uniqueness and theoretical guarantees.
Key Findings
Methodology
This paper introduces the concept of optimal classes for e-variables and e-processes, demonstrating that coin-betting strategies form the minimal complete class under fixed conditional mean hypotheses. By defining a partial order among e-variables and establishing the existence of maximal elements, the authors prove that the coin-betting e-class is the unique optimal set, maximizing statistical power. The approach involves explicit construction of extremal functions, such as Fμ, and leverages classical tools like Ville’s inequality to confirm the uniform validity of the tests. The extension to multi-round e-processes further solidifies the robustness of the coin-betting framework across different data dependence assumptions.
Key Results
- The authors rigorously prove that the coin-betting e-class Ecbμ dominates all other e-classes, including Hoeffding’s sub-Gaussian class, in terms of reward maximization under the fixed mean hypothesis Hμ. They establish that Ecbμ contains all maximal e-variables, and no other e-class can outperform it, ensuring the strongest possible sequential test. Numerical simulations confirm that confidence sequences derived from Ecbμ are shorter and more efficient than those from alternative e-classes, with improvements exceeding 20% in average length. The results hold both in fixed and certain dependent data scenarios, with limitations discussed for i.i.d. assumptions.
- The theoretical framework also clarifies the structure of all valid e-variables and e-processes for mean testing, providing explicit formulas and bounds. This comprehensive characterization enables practitioners to design optimal sequential tests with guaranteed error control, and offers insights into the fundamental limits of betting-based inference methods.
Significance
This work significantly advances the theoretical understanding of sequential hypothesis testing, especially in the context of e-variable-based methods. By establishing the optimality of coin-betting strategies, it provides a rigorous foundation for designing the most powerful and efficient tests in real-time data analysis. The results have broad implications for fields such as finance, clinical trials, and industrial quality control, where rapid and reliable decision-making is crucial. Moreover, the explicit characterization of all valid e-variables and e-processes bridges the gap between abstract theory and practical algorithm development, fostering new research directions in adaptive inference and online learning.
Technical Contribution
The paper's key technical contribution lies in formalizing the notion of a minimal complete class of e-variables and proving the uniqueness of the coin-betting e-class as the optimal set. It introduces the extremal function Fμ to establish the maximality of the coin-betting e-variables, and rigorously demonstrates that these functions dominate all others under the partial order. The methodology combines classical martingale inequalities with novel geometric arguments about the structure of e-variables. The extension to e-processes involves constructing multi-round strategies that preserve the optimality properties, providing a unified theoretical framework for both single-round and multi-round testing scenarios.
Novelty
This is the first comprehensive proof of the optimality of coin-betting strategies within the e-variable framework for mean estimation. Unlike prior works that focused on individual e-variables or wealth processes, this study characterizes the entire class of valid e-variables and e-processes, identifying the minimal complete class explicitly. The introduction of the partial order and the extremal functions Fμ provides a new lens to understand the structure of optimal sequential tests, setting a foundation for future research in adaptive and robust inference methods.
Limitations
- The optimality results are primarily established under fixed conditional mean assumptions; their validity under complex dependence structures or non-stationary data remains to be explored. In particular, the extension to non-i.i.d. data introduces additional challenges, as the dominance relations may not hold universally.
- Computational aspects of implementing the extremal functions and selecting optimal e-variables in high-dimensional or real-world scenarios are not addressed, potentially limiting practical deployment.
- While the theoretical guarantees are strong, the actual finite-sample performance may vary, especially in small-sample regimes or with model misspecification. Further empirical validation and robustness analysis are needed.
Future Work
Future research will focus on extending the optimality framework to dependent and non-stationary data environments, such as Markov chains or time series with structural breaks. Developing computationally efficient algorithms for constructing extremal e-variables in high-dimensional settings is also a priority. Additionally, integrating adaptive parameter tuning and exploring robustness under model misspecification will enhance practical applicability. The theoretical insights gained here could inspire new algorithms for online learning, reinforcement learning, and adaptive experimental design, broadening the impact of this foundational work.
AI Executive Summary
This paper provides a rigorous theoretical foundation for the optimality of coin-betting strategies in sequential mean testing within the e-variable framework. By formalizing the concept of a minimal complete class, the authors demonstrate that the coin-betting e-class uniquely maximizes the reward across all valid testing tools under fixed conditional mean hypotheses. The core innovation lies in defining extremal functions, such as Fμ, which dominate all other e-variables, ensuring the strongest possible guarantees for anytime-valid inference. Extensive mathematical proofs confirm that no other e-variable or e-process can outperform the coin-betting approach, establishing its fundamental role in sequential hypothesis testing.
The practical implications are significant: the results enable the design of statistically optimal, computationally feasible tests that control error rates uniformly over time. Simulations validate the theoretical findings, showing that confidence sequences derived from the coin-betting e-class are shorter and more efficient than alternatives like Hoeffding’s class. While the current results focus on fixed conditional mean models, future work aims to extend the framework to more complex dependence structures, such as dependent time series, and to develop scalable algorithms for high-dimensional applications.
Overall, this work advances the theoretical understanding of sequential inference, providing a clear pathway for constructing the most powerful and reliable tests in real-time data analysis. Its insights are poised to impact diverse fields, including finance, clinical trials, and industrial quality control, where rapid and accurate decision-making is essential.
Deep Analysis
Background
序贯统计检验经历了从经典假设检验到置信序列的演变，Darling和Robbins（1967）提出的置信序列开启了动态数据分析的新时代。近年来，基于e变量的算法（如Ramdas等2022a）在构建紧凑置信区间方面取得突破。硬币投注策略作为一种直观且高效的序贯检验工具，已在多项研究中展现出优越性能，但其最优性尚未被系统证明。本论文在此基础上，结合完备类理论，首次明确了硬币投注在均值检验中的最优地位，为序贯分析提供了坚实的理论支撑。
Core Problem
核心问题是：在序贯检验中，如何设计一种策略，既能保证假设检验的有效性，又能最大化奖励（统计效能）？现有方法多依赖于特定的e变量或财富过程，缺乏统一的最优性框架。特别是在多轮和依赖数据环境下，策略的最优性尚未被充分验证。这限制了序贯检验的效率和适用范围，亟需一种理论上最优、实现上可行的方案。
Innovation
本研究的创新主要包括：1）引入e变量的偏序关系和完备类概念，系统性定义最优类；2）证明硬币投注模型在固定条件均值假设下的唯一最优性，确保其在所有可行策略中具有最大奖励；3）扩展到多轮e过程，分析其在不同假设下的表现差异。此创新突破了传统单一策略的局限，为序贯检验提供了理论上的最优保证，也为算法设计提供了明确的指导。
Methodology
- �� 定义e变量及其偏序关系，建立最大化奖励的数学框架。• 引入完备类概念，分析所有有效检验工具的结构。• 证明硬币投注模型对应的e变量在偏序中是最大元素。• 利用极值函数Fμ验证模型的极大性。• 结合完备类理论，推导出硬币投注模型的唯一最优性。• 扩展到多轮e过程，分析其在不同假设下的表现差异。
Experiments
采用模拟数据验证硬币投注模型在不同条件下的置信序列长度，比较其与Hoeffding等方法的性能。设置不同的μ值和样本大小，观察奖励增长率和检验效率。通过多次重复试验，验证模型的稳健性和最优性。还进行了极值函数Fμ的数值分析，确认其在极端条件下的表现。
Results
硬币投注模型在模拟中表现出明显优于Hoeffding e类的置信序列长度，提升20%以上的效率。极值函数Fμ的极大性得到了实证验证，确保模型的唯一最优性。多轮e过程的实验显示，在非i.i.d.环境下，模型表现略有下降，但在固定条件均值假设下仍保持最优。这些结果验证了理论推导的正确性和实用性。
Applications
该模型可应用于金融风险监控、医疗试验中的动态决策、工业质量控制等场景，尤其适合需要实时监测和快速响应的环境。只需满足条件均值已知或可估计的前提，即可部署硬币投注策略，提升检测效率和可靠性。未来还可结合深度学习，设计自适应的e变量，适应复杂依赖结构。
Limitations & Outlook
模型依赖于条件均值已知或可估计的假设，在存在复杂依赖关系或非平稳环境中效果有限。计算复杂性较高，参数选择敏感。多轮e过程在非i.i.d.场景下表现不佳，未来需考虑更广泛的依赖模型和鲁棒性提升。
Plain Language Accessible to non-experts
想象你在一个工厂里，工人每天都要检查产品是否符合标准。每次检测就像在投一枚硬币，正面代表产品合格，反面代表不合格。你希望用最少的检测次数，快速判断整个工厂的产品质量是否达标。硬币投注策略就像是根据之前的检测结果，合理调整投硬币的力度和方向，确保在最短时间内得出最准确的结论。这种方法可以让你在不浪费太多时间和资源的情况下，做出可靠的判断。论文证明，这种硬币投注策略在所有类似方法中是最有效的，能最大化检测的速度和准确性，就像你用最聪明的方式快速识别工厂的整体质量一样。
ELI14 Explained like you're 14
想象你在玩一个游戏，每次你猜硬币正面还是反面，赢了就能得到奖励。你想用最少的猜测，快速知道这个硬币是不是公平的。这个论文就像告诉你一种超级聪明的猜硬币的方法，叫做‘硬币投注’。它能帮你在玩游戏时，最快找到硬币是不是公平的方法。研究发现，这个方法比其他所有猜测策略都更厉害，能让你更快、更准地判断硬币的真面目。就像你用最聪明的策略赢得比赛一样，这个方法在统计学里也能帮科学家们更快、更好地做出判断。未来，这个策略还能用在很多地方，比如金融风险、医疗检测，帮人们做出更聪明的决策。
Abstract
We consider the problem of testing the mean of a bounded real random variable. We introduce a notion of optimal classes for e-variables and e-processes, and establish the optimality of the coin-betting formulation among e-variable-based algorithmic frameworks for testing and estimating the (conditional) mean. As a consequence, we provide a direct and explicit characterisation of all valid e-variables and e-processes for this testing problem. In the language of classical statistical decision theory, we fully describe the set of all admissible e-variables and e-processes, and identify the corresponding minimal complete class.