Detecting Changes in a Poisson Process Monitored at Unequal Discrete Time Intervals

Abstract

We look at a Poisson process where the arrival rate changes at some unknown time point. We monitor this process only at certain time points. At each time point, we count the number of arrivals that happened in that time interval. In previous work, it was assumed that the time intervals were equal. Since work schedules may prevent employees to monitor the process in the evenings or on weekends, we relax this assumption to allow for monitoring at unequal time intervals. For a loss function consisting of the cost of late detection and a penalty for early stopping, we develop, using dynamic programming, the one and two steps look ahead Bayesian stopping rules. We then compare various observation schemes to determine the best model. We provide some numerical results to illustrate the effectiveness of the detection procedures.

Keywords:

• Bayesian stopping rules
• Change-point detection
• Dynamic programming
• Poisson processes
• Risk
• Uneven time intervals

Mathematics Subject Classification:

• 62L15
• 60G40

Acknowledgments

I would like to thank the Editor, Dr. Nitis Mukhopadhyay, for inviting me to write this article. I would also like to thank the referees for their helpful comments. I especially thank my advisor, Dr. Shelley Zacks, who introduced me to research and coauthored with me on previous work from which these results were extended. I also thank him for his advice and encouragement in writing this article.