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A class of weighted Poisson processes

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  • Balakrishnan, N.
  • Kozubowski, Tomasz J.

Abstract

Let N have a Poisson distribution with parameter [lambda]>0, and let U1,U2,... be a sequence of independent standard uniform variables, independent of N. Then the random sum , where IA is an indicator of the set A, is a Poisson process on [0,1]. Replacing N by its weighted version Nw, we obtain another process with weighted Poisson marginal distributions. We then derive the basic properties of such processes, which include marginal and joint distributions, stationarity of the increments, moments, and the covariance function. In particular, we show that properties of overdispersion and underdispersion of N(t) are related to the correlation of the process increments, and are equivalent to the analogous properties of Nw. Theoretical results are illustrated through examples, which include processes with geometric and negative binomial marginal distributions.

Suggested Citation

  • Balakrishnan, N. & Kozubowski, Tomasz J., 2008. "A class of weighted Poisson processes," Statistics & Probability Letters, Elsevier, vol. 78(15), pages 2346-2352, October.
    • Handle: RePEc:eee:stapro:v:78:y:2008:i:15:p:2346-2352
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    References listed on IDEAS

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    1. Joan Del Castillo & Marta Pérez-Casany, 1998. "Weighted Poisson Distributions for Overdispersion and Underdispersion Situations," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 50(3), pages 567-585, September.
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    Cited by:

    1. Macci, Claudio & Pacchiarotti, Barbara, 2015. "Large deviations for a class of counting processes and some statistical applications," Statistics & Probability Letters, Elsevier, vol. 104(C), pages 36-48.
    2. Chedly Gelin Louzayadio & Rodnellin Onesime Malouata & Michel Diafouka Koukouatikissa, 2021. "A Weighted Poisson Distribution for Underdispersed Count Data," International Journal of Statistics and Probability, Canadian Center of Science and Education, vol. 10(4), pages 157-157, July.
    3. Lefèvre, Claude & Picard, Philippe, 2011. "A new look at the homogeneous risk model," Insurance: Mathematics and Economics, Elsevier, vol. 49(3), pages 512-519.
    4. Roberto Garra & Enzo Orsingher & Federico Polito, 2018. "A Note on Hadamard Fractional Differential Equations with Varying Coefficients and Their Applications in Probability," Mathematics, MDPI, vol. 6(1), pages 1-10, January.
    5. Beghin, Luisa & Macci, Claudio, 2017. "Asymptotic results for a multivariate version of the alternative fractional Poisson process," Statistics & Probability Letters, Elsevier, vol. 129(C), pages 260-268.
    6. Leda Minkova & N. Balakrishnan, 2013. "Compound weighted Poisson distributions," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 76(4), pages 543-558, May.
    7. Beghin, Luisa & Macci, Claudio, 2013. "Large deviations for fractional Poisson processes," Statistics & Probability Letters, Elsevier, vol. 83(4), pages 1193-1202.

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