Online Bootstrap Inference for the Trend of Nonstationary Time Series

TL;DR

This paper introduces an online bootstrap method for nonparametric trend estimation in nonstationary time series, enabling scalable and reliable uncertainty quantification for streaming data applications.

Contribution

It presents a novel online bootstrap scheme applicable to a broad class of level estimators, providing uniform-in-time coverage and practical inference in nonstationary streaming contexts.

Findings

Good finite-sample performance demonstrated in simulations

Applicable to exponential smoothing and moving averages

Enables scalable uncertainty quantification in streaming data

Abstract

This article proposes an online bootstrap scheme for nonparametric level estimation in nonstationary time series. Our approach applies to a broad class of level estimators expressible as weighted sample averages over time windows, including exponential smoothing methods and moving averages. The bootstrap procedure is motivated by asymptotic arguments and provides well-calibrated uniform-in-time coverage, enabling scalable uncertainty quantification in streaming or large-scale time-series settings. This makes the method suitable for tasks such as adaptive anomaly detection, online monitoring, or streaming A/B testing. Simulation studies demonstrate good finite-sample performance of our method across a range of nonstationary scenarios. In summary, this offers a practical resampling framework that complements online trend estimation with reliable statistical inference.