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Entropy
Volume 27
Issue 5
10.3390/e27050537
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Article

Bootstrap Confidence Intervals for Multiple Change Points Based on Two-Stage Procedures

by
Li Hou
1,
Baisuo Jin
1,*,
Yuehua Wu
2,* and
Fangwei Wang
1
1
Department of Statistics and Finance, University of Science and Technology of China, Hefei 230026, China
2
Department of Mathematics and Statistics, York University, Toronto, ON M3J 1P3, Canada
*
Authors to whom correspondence should be addressed.
Entropy 2025, 27(5), 537; https://doi.org/10.3390/e27050537 (registering DOI)
Submission received: 15 April 2025 / Revised: 12 May 2025 / Accepted: 15 May 2025 / Published: 17 May 2025
(This article belongs to the Special Issue Statistical Methods for Modeling High-Dimensional and Complex Data: Second Edition)
Versions Notes

Abstract

This paper investigates the construction of confidence intervals for multiple change points in linear regression models. First, we detect multiple change points by performing variable selection on blocks of the input sequence; second, we re-estimate their exact locations in a refinement step. Specifically, we exploit an orthogonal greedy algorithm to recover the number of change points consistently in the cutting stage, and employ the sup-Wald-type test statistic to determine the locations of multiple change points in the refinement stage. Based on a two-stage procedure, we propose bootstrapping the estimated centered error sequence, which can accommodate unknown magnitudes of changes and ensure the asymptotic validity of the proposed bootstrapping method. This enables us to construct confidence intervals using the empirical distribution of the resampled data. The proposed method is illustrated with simulations and real data examples.
Keywords: change point; bootstrap; confidence interval; OGA; sup-Wald-type test change point; bootstrap; confidence interval; OGA; sup-Wald-type test

Share and Cite

MDPI and ACS Style

Hou, L.; Jin, B.; Wu, Y.; Wang, F. Bootstrap Confidence Intervals for Multiple Change Points Based on Two-Stage Procedures. Entropy 2025, 27, 537. https://doi.org/10.3390/e27050537

AMA Style

Hou L, Jin B, Wu Y, Wang F. Bootstrap Confidence Intervals for Multiple Change Points Based on Two-Stage Procedures. Entropy. 2025; 27(5):537. https://doi.org/10.3390/e27050537

Chicago/Turabian Style

Hou, Li, Baisuo Jin, Yuehua Wu, and Fangwei Wang. 2025. "Bootstrap Confidence Intervals for Multiple Change Points Based on Two-Stage Procedures" Entropy 27, no. 5: 537. https://doi.org/10.3390/e27050537

APA Style

Hou, L., Jin, B., Wu, Y., & Wang, F. (2025). Bootstrap Confidence Intervals for Multiple Change Points Based on Two-Stage Procedures. Entropy, 27(5), 537. https://doi.org/10.3390/e27050537

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