Sequential Detection and Estimation of Change-Points

Abstract

The problem of sequential detection and estimation of a change-point is considered. The ‘large parameter’ approach for solving this problem is proposed. A two-stage method is designed according to this approach, which includes the nonparametric version of the CUSUM procedure for sequential change-point detection and the modified Kolmogorov-Smirnov test for the retrospective estimation of the detected change-point. The asymptotic effectiveness of this method is proved for dependent observations (Theorems 2.1 and 3.1) when the large parameter infinitely increases. Monte Carlo study of this method for the Gaussian case (changes in mean and dispersion) is performed. The a priori theoretical lower bounds are proved for new performance measures in sequential change-point detection and retrospective estimation (Theorems 5.1 and 5.2) and the asymptotic optimality of the proposed method is demonstrated.

Keywords:

• Asymptotic optimality
• Change-point
• Change-point estimation
• Performance measure
• Sequential detection

Subject Classifications:

• 62L10

ACKNOWLEDGMENTS

This paper was invited and presented during the Second International Workshop in Sequential Methodologies, June 15–17, 2009, held in Troyes, France. I am grateful to Prof. A. Tartakovsky, Prof. A. Shiryaev, and Prof. N. Mukhopadhyay for many valuable comments and suggestions, which helped to improve the draft of this paper.

Notes

Recommended by N. Mukhopadhyay