Stopping Rules for Monte Carlo Methods: A Review

TL;DR

This review paper comprehensively discusses recent advances in sequential stopping rules for Monte Carlo methods, focusing on iid sampling and their applications in statistical inference.

Contribution

It provides an up-to-date synthesis of theoretical and practical developments in sequential stopping rules for Monte Carlo estimation, emphasizing core assumptions and algorithms.

Findings

Reviewed over a hundred references and empirical studies.

Analyzed core assumptions, algorithms, and convergence properties.

Highlighted practical trade-offs and future research directions.

Abstract

Sequential analysis encompasses simulation theories and methods where the sample size is determined dynamically based on accumulating data. Since the conceptual inception, numerous sequential stopping rules have been introduced, and many more are currently being refined and developed. This article aims to deliver a comprehensive and up-to-date review of recent developments on sequential stopping rules, intentionally emphasizing standard iid Monte Carlo methods and lightly generalized ones, employed primarily for estimating an unknown expectation, including binomial proportions. These methodologies have long served and likely will continue to serve, as fundamental bases for both theoretical and practical developments in stopping rules for general statistical inference, advanced Monte Carlo techniques and their modern applications. Building upon over a hundred references and empirical studies, we explore the essential aspects of these methods, such as core assumptions, numerical algorithms, convergence properties, and practical trade-offs to guide further developments, particularly at the intersection of sequential stopping rules and related areas of research.