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Online Bootstrap Inference for the Trend of Nonstationary Time Series

Thomas Nagler, Tobias Brock, Nicolai Palm

TL;DR

An online bootstrap scheme for nonparametric level estimation in nonstationary time series is proposed, offering a practical resampling framework that complements online trend estimation with reliable statistical inference.

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.

Online Bootstrap Inference for the Trend of Nonstationary Time Series

TL;DR

An online bootstrap scheme for nonparametric level estimation in nonstationary time series is proposed, offering a practical resampling framework that complements online trend estimation with reliable statistical inference.

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.
Paper Structure (53 sections, 15 theorems, 130 equations, 11 figures, 1 algorithm)

This paper contains 53 sections, 15 theorems, 130 equations, 11 figures, 1 algorithm.

Table of Contents

  1. INTRODUCTION
  2. Motivation
  3. Contribution
  4. Related Work
  5. Confidence sequences/sequential monitoring
  6. Time Series Bootstrap
  7. Adaptive Conformal Inference
  8. BACKGROUND
  9. Setup
  10. Online Estimators of the Trend
  11. The Effective Sample Size
  12. PROBLEM FORMULATION
  13. Uniform-in-time inference
  14. Confidence Bands
  15. Testing
  16. ...and 38 more sections

Key Result

Theorem 5.1

Let $c_{t, \alpha}^* = \sigma_t^* q_{k, \alpha}^*$, $\sigma_t^{*2} = {\mathds{V}\mathrm{ar}}[\widehat{\delta}_{\eta}^{*}(t)]$, and suppose assumptions A1:X, A2:X, A0:weights, and A5:weights hold. If it holds that

Figures (11)

  • Figure 1: Simulated data, true and estimated smooth mean, and confidence bands obtained from Algorithm \ref{['algo:main']} with $\alpha=0.1$ and EWMA weigths.
  • Figure 2: Uniform coverage for various DGPs and $\alpha=0.1$ using EWMA.
  • Figure 3: Power for detecting the non-constant trends in the Trend + seasonality DGP under varying trend slopes $a$ using EWMA; $a = 0.001$ is shown in \ref{['fig:wide-compare-processes']}.
  • Figure 4: Average interval width for various stationary processes.
  • Figure 5: Uniform coverage for various DGPs and $\alpha=0.2$ using EWMA.
  • ...and 6 more figures

Theorems & Definitions (34)

  • Example 2.1: EWMA
  • Example 2.2: Brown double exponential smoothing
  • Example 2.3: Triple exponential smoothing with additive seasonality
  • Example 3.1: Jump detection
  • Theorem 5.1
  • Definition B.1
  • Lemma B.2
  • proof
  • Lemma B.3: Verifying assumptions for EWMA weights
  • proof
  • ...and 24 more