Change-Point Estimation and Detection for Mixture of Linear Regression Models

This paper studies the estimation and detection problems in the mixture of linear regression models with change point. An improved Expectation–Maximization (EM) algorithm is devised specifically for multi-classified mixture data with change points. Under appropriate conditions, the large-sample properties of the estimator are rigorously proven. This improved EM algorithm not only precisely locates the change points but also yields accurate parameter estimates for each class. Additionally, a detector grounded in the score function is proposed to identify the presence of change points in mixture data. The limiting distributions of the test statistics under both the null and alternative hypotheses are systematically derived. Extensive simulation experiments are conducted to assess the effectiveness of the proposed method, and comparative analyses with the conventional EM algorithm are performed. The results clearly demonstrate that the EM algorithm without considering change points exhibits poor performance in classifying data, often resulting in the misclassification or even omission of certain classes. In contrast, the estimation method introduced in this study showcases remarkable accuracy and robustness, with favorable empirical sizes and powers.