Sojourning With the Homogeneous Poisson Process

ABSTRACT

In this pedagogical article, distributional properties, some surprising, pertaining to the homogeneous Poisson process (HPP), when observed over a possibly random window, are presented. Properties of the gap-time that covered the termination time and the correlations among gap-times of the observed events are obtained. Inference procedures, such as estimation and model validation, based on event occurrence data over the observation window, are also presented. We envision that through the results in this article, a better appreciation of the subtleties involved in the modeling and analysis of recurrent events data will ensue, since the HPP is arguably one of the simplest among recurrent event models. In addition, the use of the theorem of total probability, Bayes’ theorem, the iterated rules of expectation, variance and covariance, and the renewal equation could be illustrative when teaching distribution theory, mathematical statistics, and stochastic processes at both the undergraduate and graduate levels. This article is targeted toward both instructors and students.

KEYWORDS:

• California earthquakes
• Iterated expectation, variance, and covariance rules
• HPP model validation
• Normalized spacings statistics
• Renewal process
• Renewal function
• Size-biased sampling
• Sum-quota accrual scheme
• Teaching statistics

Acknowledgments

The authors are very grateful to the referees, associate editor, and the editor, Professor Nicole Lazar, for their thorough reading of the manuscript and their comments, suggestions, and criticisms which helped in considerably improving the article. The authors also thank Professor James Lynch, Dr. AKM Fazlur Rahman, and graduate students Taeho Kim, Bereket Kindo, Beidi Qiang, Shiwen Shen, Jeff Thompson, Lillian Wanda, and Lu Wang for their comments, insights, and feedbacks on the article.

Funding

This research was partially supported by NSF Grant DMS1106435 and NIH Grants R01CA154731 and P30GM103336-01A1.