On the Moment Characteristics for the Univariate Compound Poisson and Bivariate Compound Poisson Processes with Applications

Sobre las características de los momentos de los procesos de Poisson compuestos univariados y bivariados con aplicaciones

The univariate and bivariate compound Poisson process (CPP and BCPP, respectively) ensure a better description than the homogeneous Poisson process for clustering of events. In this paper, new explicit representations of the moment characteristics (general, central, factorial, binomial and ordinary moments, factorial cumulants) and some covariance structures are derived for the CPP and BCPP. Then, the skewness and kurtosis of the univariateCPP are obtained for the first time and special cases of the CPP are studied in detail. Applications to two real data sets are given to illustrate the usage of these processes.

Los procesos univariados y bivariados compuestos de Poisson (CPP y BCCPP, por sus siglas en inglés respectivamente) permiten una mejor descripción que los procesos homogéneos de Poisson para agrupamiento de eventos. En este artículo, se muestran específicamente las representaciones de las características de momentos (general, central, factorial, momentos binomiales y ordinarios, acumuladas factoriales) y algunas estructuras de covarianza para los CPP y BCPP. Adicionalmente, el sesgo y la curtosis de los procesos univariados CPP son presentados y casos especiales son estudiados en detalle. La aplicación a dos conjuntos de datos reales es usada con el fin de ilustrar el uso de estos procesos.

On the Moment Characteristics for the Univariate Compound Poisson and Bivariate Compound Poisson Processes with Applications Sobre las características de los momentos de los procesos de Poisson compuestos univariados y bivariados con aplicaciones GAMZE ÖZEL¹

¹Hacettepe University, The Faculty of Science, Department of Statistics, Ankara, Turkey. Lecturer. Email: gamzeozl@hacettepe.edu.tr

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Abstract

The univariate and bivariate compound Poisson process (CPP and BCPP, respectively) ensure a better description than the homogeneous Poisson process for clustering of events. In this paper, new explicit representations of the moment characteristics (general, central, factorial, binomial and ordinary moments, factorial cumulants) and some covariance structures are derived for the CPP and BCPP. Then, the skewness and kurtosis of the univariate CPP are obtained for the first time and special cases of the CPP are studied in detail. Applications to two real data sets are given to illustrate the usage of these processes.

Key words: Bivariate distribution, Compound Poisson process, Cumulant, Factorial moments, Moment.

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Resumen

Palabras clave: acumuladas factoriales, conjuntas, distribución bivariada, distribución compuesta de Poisson, momento.

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Texto completo disponible en PDF

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References

1. Agresti, A. (2002), Categorical Data Analysis, John Wiley & Sons, New Jersey.

2. Ata, N. & Özel, G. (2012), 'Survival functions for the frailty models based on the discrete compound Poisson process', Journal of Statistical Computation and Simulation (Online Published). DOI: 10.1080/00949655.2012.679943.

3. Chen, C. W., Randolph, P. & Tian-Shy, L. (2005), 'Using CUSUM control schemes for monitoring quality levels in compound Poisson production environments: the geometric Poisson process', Quality Engineering 17, 207-217.

4. Christophersen, A. & Smith, E. G. C. (2000), A global model for aftershock behaviour, '', Proceedings of the 12th World Conference on Earthquake Engineering. Paper 0379, Auckland, New Zealand.

5. Getis, A. (1974), Representation of spatial point processes by Pólya methods, '', Proceedings of the 1972 meeting of the IGU Commission on Quantitative Geography. Montreal, Canada.

6. Gudowska-Nowak, E., Lee, R., Nasonova, E., Ritter, S. & Scholz, M. (2007), 'Effect of let and track structure on the statistical distribution of chromosome aberrations', Advances in Space Research 39, 1070-1075.

7. Hesselager, O. (1996), 'Recursions for certain bivariate counting distributions and their compound distributions', ASTIN Bulletin 26, 35-52.

8. Kocherlakota, S. & Kocherlakota, K. (1997), Bivariate Discrete Distributions, Wiley, NewYork.

9. Meintanis, S. G. (1997), 'A new goodness of fit test for certain bivariate distributions applicable to traffic accidents', Statistical Methodology 4, 22-34.

10. Neyman, J. (1939), 'On a new class of contagious distributions applicable in entomology and bacteriology', Annals of Mathematical Statistics 10, 35-57.

11. Robin, S. (2002), 'A compound Poisson model for word occurrences in DNA sequences', Applied Statistics 51, 437-451.

12. Rosychuk, R. J., Huston, C. & Prasad, N. G. N. (2006), 'Spatial event cluster detection using a compound Poisson distribution', Biometrics 62, 465-470.

13. Sundt, B. (1992), 'On some extensions of Panjer's class of counting distributions', ASTIN Bulletin 22, 61-80.

14. Wienke, A. (2011), Frailty Model in Survival Analysis, Chapman and Hall.

15. Wienke, A., Ripatti, S., Palmgren, J. & Yashin, A. (2010), 'A bivariate survival model with compound Poisson frailty', Statistics in Medicine 29(2), 275-283.

16. Özel, G. (2011a), 'On certain properties of a class of bivariate compound Poisson distributions and an application to earthquake data', Revista Colombiana de Estadistica 34(3), 545-566.

17. Özel, G. (2011b), 'A bivariate compound Poisson model for the occurrence of foreshock and aftershock sequences in Turkey', Environmetrics 22(7), 847-856.

18. Özel, G. & Inal, C. (2008), 'The probability function of the compound Poisson process and an application to aftershock sequences', Environmetrics 19, 79-85.

19. Özel, G. & Inal, C. (2010), 'The probability function of a geometric Poisson distribution', Journal of Statistical Computation and Simulation 80, 479-487.

20. Özel, G. & Inal, C. (2012), 'On the probability function of the first exit time for generalized Poisson processes', Pakistan Journal of Statistics 28(1), 27-40.

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[Recibido en marzo de 2012. Aceptado en marzo de 2013]

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Este artículo se puede citar en LaTeX utilizando la siguiente referencia bibliográfica de BibTeX:

@ARTICLE{RCEv36n1a04,
    AUTHOR  = {Özel, Gamze},
    TITLE   = {{On the Moment Characteristics for the Univariate Compound Poisson and Bivariate Compound Poisson Processes with Applications}},
    JOURNAL = {Revista Colombiana de Estadística},
    YEAR    = {2013},
    volume  = {36},
    number  = {1},
    pages   = {59-77}
}

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