Nonparametric Shiryaev-Roberts change-point detection procedures based on modified empirical likelihood

Abstract

Sequential change-point analysis, which identifies a change of probability distribution in an infinite sequence of random observations, has important applications in many fields. A good method should detect a change point as soon as possible, and keep a low amount of false alarms. As one of the most popular methods, Shiryaev-Roberts (SR) procedure holds many optimalities. However, its implementation requires the pre-change and post-change distributions to be known, which is not achievable in practice. In this paper, we construct a nonparametric version of the SR procedure by embedding different versions of empirical likelihood, assuming two training samples, before and after change, are available for parameter estimations. Simulations are conducted to compare the performance of the proposed method with existing methods. The results show that when the underlying distribution is unknown, and training sample sizes are small, the proposed modified procedure shows advantage by giving a smaller delay of detection.

Keywords:

• Sequential change-point analysis
• Shiryaev-Roberts procedure
• stopping time
• empirical likelihood
• online monitoring

Acknowledgments

The authors are grateful to the Editor-in-chief, Dr. Jie Chen, an associate editor, and two anonymous reviewers for their helpful comments on improving the contents of this paper.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Data and code availability

R code implementing the methods proposed in this paper is available from https://github.com/peiyao2017/EL-SR. R-4.2.1 is available from https://cloud.r-project.org/. The aircraft data that support the findings of this study are available from https://github.com/peiyao2017/EL-SR. The paper containing raw data is available at https://www.jstor.org/stable/1266340.

Notes

1 R code implementing the methods proposed in this paper is available from https://github.com/peiyao2017/EL-SR. The results when $AR{L}_0\approx 1000$ and m = 20 are obtained for uniform and Poisson distributions. We also provide the false alarm probabilities (FAP) in each table to give a more complete view of the performance of the methods, as suggested by the reviewer.

Additional information

Funding

Ning's work is supported by the Simons Foundation [grant number 711800].