OPTICS: Order-Preserved Test-Inverse Confidence Set for Number of Change-Points

2026-03-29

AI Summary

Existing methods for estimating the number of change points typically provide only point estimates, lacking statistical guarantees for the true number of change points and relying on strong assumptions that are difficult to verify. This work proposes a unified framework based on test inversion and a multi-split strategy, which, for the first time, constructs a confidence set for the number of change points with rigorous theoretical guarantees. The approach enables valid inference under substantially weaker conditions and offers a principled means to evaluate the performance of existing estimators. It accommodates heavy-tailed and dependent data, and theoretical analysis controls the order of the confidence set size. Numerical experiments demonstrate its computational efficiency and informativeness, while its practical utility and superiority are further confirmed through an analysis of bladder tumor microarray data.

Abstract

Determining the number of change-points is a first-step and fundamental task in change-point detection problems, as it lays the groundwork for subsequent change-point position estimation. While the existing literature offers various methods for consistently estimating the number of change-points, these methods typically yield a single point estimate without any assurance that it recovers the true number of changes in a specific dataset. Moreover, achieving consistency often hinges on very stringent conditions that can be challenging to verify in practice. To address these issues, we introduce a unified test-inverse procedure to construct a confidence set for the number of change-points. The proposed confidence set provides a set of possible values within which the true number of change-points is guaranteed to lie with a specified level of confidence. We further proved that the confidence set is sufficiently narrow to be powerful and informative by deriving the order of its cardinality. Remarkably, this confidence set can be established under more relaxed conditions than those required by most point estimation techniques. We also advocate multiple-splitting procedures to enhance stability and extend the proposed method to heavy-tailed and dependent settings. As a byproduct, we may also leverage this constructed confidence set to assess the effectiveness of point-estimation algorithms. Through extensive simulation studies, we demonstrate the superior performance of our confidence set approach. Additionally, we apply this method to analyze a bladder tumor microarray dataset. Supplementary Material, including proofs of all theoretical results, computer code, the R package, and extended simulation studies, are available online.

Problem

• change-point detection
• number of change-points
• confidence set
• statistical guarantee
• consistency

Innovation

• change-point detection
• confidence set
• test-inversion
• multiple splitting
• heavy-tailed data