TIME-UNIFORM CENTRAL LIMIT THEORY AND ASYMPTOTIC CONFIDENCE SEQUENCES

• 成果类型: Article
• 署名作者: Waudby-smith, Ian; Arbour, David; Sinha, Ritwik; Kennedy, Edward H.; Ramdas, Aaditya
• 署名单位: Carnegie Mellon University; Adobe Systems Inc.
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• DOI: 10.1214/24-AOS2408
• 发表日期: 2024
• 页码: 2613-2640
• 关键词: boundary crossing probabilities; partial sums; approximation; regression

摘要

Confidence intervals based on the central limit theorem (CLT) are a cornerstone of classical statistics. Despite being only asymptotically valid, they are ubiquitous because they permit statistical inference under weak assumptions and can often be applied to problems even when nonasymptotic inference is impossible. This paper introduces time-uniform analogues of such asymptotic confidence intervals, adding to the literature on confidence sequences (CS)-sequences of confidence intervals that are uniformly valid over time-which provide valid inference at arbitrary stopping times and incur no penalties for peeking at the data, unlike classical confidence intervals which require the sample size to be fixed in advance. Existing CSs in the literature are nonasymptotic, enjoying finite-sample guarantees but not the aforementioned broad applicability of asymptotic confidence intervals. This work provides a definition for asymptotic CSs and a general recipe for deriving them. Asymptotic CSs forgo nonasymptotic validity for CLT-like versatility and (asymptotic) time-uniform guarantees. While the CLT approximates the distribution of a sample average by that of a Gaussian for a fixed sample size, we use strong invariance principles (stemming from the seminal 1960s work of Strassen) to uniformly approximate the entire sample average process by an implicit Gaussian process. As an illustration, we derive asymptotic CSs for the average treatment effect in observational studies (for which nonasymptotic bounds are essentially impossible to derive even in the fixedtime regime) as well as randomized experiments, enabling causal inference in sequential environments.