OPTICS: Order-Preserved Test-Inverse Confidence Set for Number of Change-Points
提出一种保序检验逆置信集方法，为变点数量提供置信区间，在更宽松条件下保证覆盖真实变点数，并可用于评估点估计算法效果。
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 propose to construct a unified order-preserved test-inverse confidence set (OPTICS) for the number of change-points. It 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 OPTICS is sufficiently narrow to be powerful and informative by deriving the order of its cardinality. Remarkably, the theory of OPTICS is established under more relaxed conditions than those required by most point estimation techniques. We also advocate multiple-splitting procedures to enhance its stability, and extend it to heavy-tailed and dependent settings. As a byproduct, we may leverage the constructed confidence set to assess the effectiveness of point-estimation algorithms. Through extensive simulation studies, we demonstrate the superior performance of OPTICS. Additionally, we apply this method to analyze a bladder tumor microarray dataset.