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Summary
This summary is machine-generated.

This study explores the homogeneous Poisson process (HPP) observed over random windows, detailing surprising distributional properties of gap-times and event correlations. It also presents inference methods for recurrent event data analysis.

Keywords:
California earthquakesHPP Model ValidationIterated Expectation, Variance, and Covariance RulesNormalized Spacings StatisticsRenewal FunctionRenewal ProcessSize-Biased SamplingSum-Quota Accrual SchemeTeaching Statistics

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Area of Science:

  • Statistics
  • Probability Theory
  • Stochastic Processes

Background:

  • The homogeneous Poisson process (HPP) is a fundamental model for recurrent events.
  • Understanding HPP behavior under random observation windows is crucial for accurate data analysis.
  • Existing literature often simplifies observation windows, potentially missing key distributional properties.

Purpose of the Study:

  • To present surprising distributional properties of the HPP observed over random windows.
  • To derive properties of gap-times and correlations among observed events.
  • To introduce inference procedures for HPP data analysis in this context.

Main Methods:

  • Analysis of distributional properties of gap-times.
  • Calculation of correlations among gap-times for observed events.
  • Development of estimation and model validation techniques based on event occurrence data.
  • Application of probability theorems (total probability, Bayes theorem) and renewal equations.

Main Results:

  • Identified novel distributional properties of the HPP under random observation.
  • Characterized gap-time properties, including those covering the termination time.
  • Established correlations among gap-times of observed events.
  • Provided practical inference procedures for analyzing HPP data.

Conclusions:

  • The study reveals complexities in HPP modeling and analysis, even for this simple model.
  • The findings enhance the appreciation of subtleties in recurrent event data.
  • The methods and results are valuable for teaching stochastic processes and mathematical statistics.