Communications for Statistical Applications and Methods

The Korean Statistical Society (한국통계학회)
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Charlier series approximation for nonhomogenious Poisson processes

  • Hyung-Tae Ha (Department of Applied Statistics, Gachon University)
  • Received : 2024.07.05
  • Accepted : 2024.08.18
  • Published : 2024.11.30
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Abstract

This study investigates the Charlier series approximation for modeling nonhomogeneous Poisson processes. It focuses on mixtures of Poisson distributions and Markov-Modulated Poisson processes to address complex temporal data patterns, such as hospital admission rates. The Charlier series approximation is constructed by expanding probability mass functions using Charlier orthogonal polynomials, which allow for adjustments to reflect higher-order moments like skewness and kurtosis. These polynomials are combined with a Poisson weight function to create flexible approximations tailored to the variability in event rates. Two artificial examples demonstrate the method's effectiveness in capturing dynamic event behaviors. A real-world application to hospital admission data further highlights its practical utility. Performance is assessed using Kullback-Leibler divergence, quantifying the improvement over simple Poisson models. The results show that the Charlier series provides enhanced data fitting and deeper insights into complex probabilistic structures.

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Acknowledgement

This research was supported by the MSIT (Ministry of Science and ICT), Korea, under the ITRC (Information Technology Research Center) support program (RS-2023-00259004) supervised by the IITP (Institute for Information and Communications Technology Planning and Evaluation).

References

  1. Barbour AD, Holst L, and Janson S (1992). Poisson Approximation, Oxford University Press, New York.
  2. Billingsley P (1995). Probability and Measure, John Wiley and Sons, New York.
  3. Fokianos K, Rahbek A, and Tjostheim D (2009). Poisson autoregression, Journal of the American Statistical Association, 104, 1430-1439.
  4. Glynn PW and Iglehart DL (1990). Simulation output analysis using standardized time series, Mathematics of Operations Research, 15, 1-16.
  5. Grandell J (1997). Mixed Poisson Processes, Chapman and Hall/CRC, London.
  6. Ha H-T and Provost SB (2007). A viable alternative to resorting to statistical tables, Communications in Statistics-Simulation and Computation, 36, 1135-1151.
  7. Hall P and Heyde CC (2014). Martingale Limit Theory and Its Application, Academic Press, New York.
  8. Johnson NL, Kemp AW, and Kotz S (2005). Univariate Discrete Distributions, John Wiley and Sons, Hoboken, New Jersey.
  9. Karlis D (2005). EM algorithm for mixed Poisson and other discrete distributions, Astin Bulletin, 35, 3-24.
  10. Kokonendji C, Senga Kiesse A, and Zocchi B (2010). Discrete distributions in statistical modelling ' for insurance data, Insurance: Mathematics and Economics, 47, 217-225.
  11. Lehmann EL and Casella G (1998). Theory of Point Estimation, Springer, New York.
  12. Provost SB and Ha H-T (2015). Distribution approximation and modeling via orthogonal polynomial sequences, Statistics, 50, 454-470.
  13. Shorack GR and Wellner JA (2009). Empirical Processes with Applications to Statistics, SIAM, Philadelphia, Pa.
  14. Stuart A and Ord JK (1994). Kendall's Advanced Theory of Statistics, Volume 1: Distribution Theory, Wiley.
  15. Sundt R and Vernic A (2009). Recursions for Convolutions and Compound Distributions with Insurance Applications, Springer, Dordrecht.