A Likelihood Ratio Approach to Sequential Change Point Detection for a General Class of Parameters

Abstract

In this article, we propose a new approach for sequential monitoring of a general class of parameters of a d-dimensional time series, which can be estimated by approximately linear functionals of the empirical distribution function. We consider a closed-end method, which is motivated by the likelihood ratio test principle and compare the new method with two alternative procedures. We also incorporate self-normalization such that estimation of the long-run variance is not necessary. We prove that for a large class of testing problems the new detection scheme has asymptotic level α and is consistent. The asymptotic theory is illustrated for the important cases of monitoring a change in the mean, variance, and correlation. By means of a simulation study it is demonstrated that the new test performs better than the currently available procedures for these problems. Finally, the methodology is illustrated by a small data example investigating index prices from the dot-com bubble. Supplementary materials for this article are available online.

Keywords:

• Change point analysis
• Likelihood ratio principle
• Self-normalization
• Sequential monitoring

Acknowledgments

The authors would like to thank Claudia Kirch, Dominik Wied, and Wei Biao Wu for some helpful discussions on this subject. We are also grateful to the three unknown referees and the associate editor for their constructive comments on an earlier version of the article.

Additional information

Funding

This work has been supported in part by the Collaborative Research Center “Statistical modeling of nonlinear dynamic processes” (SFB 823, Teilprojekt A1, C1) and the Research Training Group “High-dimensional phenomena in probability—fluctuations and discontinuity” (RTG 2131) of the German Research Foundation (DFG).