[Paper Review] Time-uniform central limit theory and asymptotic confidence sequences

Ian Waudby-Smith, David Arbour+3 more|arXiv (Cornell University)|Mar 11, 2021

Probability and Risk Models4 citations

TL;DR

This paper introduces asymptotic confidence sequences (AsympCSs), a novel time-uniform framework that extends classical central limit theorem (CLT) inference to sequential settings. By leveraging strong invariance principles from Strassen and Komlós-Major-Tusnády, it constructs universally valid confidence sequences under the same weak moment conditions as the CLT, enabling continuous monitoring and adaptive stopping for causal inference in observational and randomized studies without sacrificing asymptotic efficiency.

ABSTRACT

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 fixed-time regime) as well as randomized experiments, enabling causal inference in sequential environments.

Motivation & Objective

• To bridge the gap between asymptotic confidence intervals and time-uniform inference by defining asymptotic confidence sequences (AsympCSs).
• To develop a universal AsympCS that maintains validity under the same weak moment assumptions required by the classical central limit theorem.
• To enable continuous, data-adaptive monitoring and inference in causal inference problems—especially in observational studies—where nonasymptotic bounds are unattainable.
• To provide a framework that combines the broad applicability of asymptotic inference with the flexibility of confidence sequences, avoiding penalties for peeking or early stopping.

Proposed method

• Uses strong invariance principles (Strassen, Komlós-Major-Tusnády) to uniformly approximate the sample average process by a Gaussian process over time.
• Applies Robbins' normal mixture boundary to construct time-uniform confidence sequences with exact type-I error control.
• Employs sequential sample splitting and cross-fitting to estimate nuisance functions in semiparametric models while maintaining asymptotic validity.
• Derives asymptotic confidence sequences for average treatment effects using efficient influence functions, with variance estimation via $σ^2_t = \widehat{\mathrm{var}}_t(\widehat{f})$.
• Establishes that the width of the AsympCS scales with the estimated standard deviation of the efficient influence function, ensuring variance-adaptivity.
• Proves that the resulting AsympCS is unimprovable in key components: the normal mixture boundary, approximation error rate, and standard error estimation.

Experimental results

Research questions

• RQ1 Can a time-uniform confidence sequence be constructed that is asymptotically valid under the same weak moment conditions as the classical central limit theorem?
• RQ2 Is it possible to construct an asymptotic confidence sequence that maintains validity under data-dependent stopping times without requiring nonasymptotic bounds?
• RQ3 How can asymptotic inference be extended to sequential settings in causal inference, especially in observational studies where nonasymptotic bounds do not exist?
• RQ4 Can the width of the confidence sequence be adaptively scaled by the estimated variance, as in classical CLT intervals, while preserving time-uniform coverage?
• RQ5 Are there fundamental limits to tightening the confidence sequence width, and is the proposed construction optimal in a minimax sense?

Key findings

• The proposed asymptotic confidence sequence is universally valid under the same moment conditions required by the classical central limit theorem, enabling broad applicability.
• The confidence sequence width scales with the estimated standard deviation of the efficient influence function, achieving variance-adaptivity even in nonasymptotic settings.
• The method achieves time-uniform coverage without requiring fixed sample sizes, allowing continuous monitoring and adaptive stopping with no type-I error inflation.
• The construction is unimprovable in its core components: the normal mixture boundary, the approximation error rate, and the standard error estimation, due to fundamental limits from strong invariance principles.
• The framework enables causal inference for average treatment effects in both observational and randomized studies under sequential monitoring, where nonasymptotic bounds are impossible.
• The method inherits optimality from Robbins' normal mixture boundary and the almost-sure approximation of sample averages by Brownian motion, making it minimax optimal in a sequential setting.