Inference for change point and post change means after a CUSUM test.
The goal of the author is to provide some quantitative evaluations on the statistical properties of estimators for change-point and post-change parameters after a signal of change is made. Most discussions are concentrated on the CUSUM procedure. At the beginning, the regular CUSUM procedure is introduced and a simple approximation for the average run lengths for design purposes is given. Then the strong renewal theorem in the exponential family case is explained. In Chapter 2, the bias and the absolute bias for the estimator of the change-point conditioning on a change is considered. The asymptotic quasi-stationary bias is derived. Then a method of constructing a lower confidence limit for the change-point is proposed. Chapter 4 concentrates on the inference problem for the post-change mean in the normal case. In Chapter 5, the behavior of the post-change mean estimator is studied when the signal is false. Then the results are extended to the problem in the normal case when the variance is subject to possible changes as well. After considering a sequential classification and segmentation procedure for alternatively changing means, an adaptive CUSUM procedure is proposed in Chapter 8 in order to deal with more complex post-change parameters. Then the methods are generalized to the correlated data case. Chapter 10 considers the estimator for the change-point under the Shiryayev-Roberts procedure. A comparison with the CUSUM procedure is conducted. The presentation is formal, all results are presented with proofs. Additionally, several case studies are included. Alltogether, readers with basic statistical training in sequential analysis with an elementary probability background are supposed to be able to understand most results.
scientific article; zbMATH DE number 2152351