In this paper, we study the estimation and inference of change points under a functional linear regression model with changes in the slope function. We present a novel Functional Regression Binary Segmentation (FRBS) algorithm which is computationally efficient as well as achieving consistency in multiple change point detection. This algorithm utilizes the predictive power of piece-wise constant functional linear regression models in the reproducing kernel Hilbert space framework. We further propose a refinement step that improves the localization rate of the initial estimator output by FRBS, and derive asymptotic distributions of the refined estimators for two different regimes determined by the magnitude of a change. To facilitate the construction of confidence intervals for underlying change points based on the limiting distribution, we propose a consistent block-type long-run variance estimator. Our theoretical justifications for the proposed approach accommodate temporal dependence and heavy-tailedness in both the functional covariates and the measurement errors. Empirical effectiveness of our methodology is demonstrated through extensive simulation studies and an application to the Standard and Poor's 500 index dataset.

翻译：本文研究斜率函数变化下函数型线性回归模型中变点的估计与推断问题。我们提出一种新颖的函数型回归二元分割（FRBS）算法，该算法不仅计算高效，而且能在多个变点检测中实现一致性。该算法利用再生核希尔伯特空间框架下分段常数函数型线性回归模型的预测能力。我们进一步提出优化步骤，以提高FRBS初始估计量的定位精度，并推导出由变化幅度决定的两个不同机制下优化估计量的渐近分布。为基于极限分布构建潜在变点的置信区间，我们提出一种一致性的块型长程方差估计量。本文方法在理论层面支持函数型协变量与测量误差的时间相依性和重尾性。通过大量仿真研究及标准普尔500指数数据集的应用，验证了该方法在实证中的有效性。