1Hacettepe University, Department of Statistics, Ankara, Turkey. Doctor. Email: gamzeozl@hacettepe.edu.tr
]]> Abstract
The univariate compound Poisson distribution has many applications in various areas such as biology, seismology, risk theory, forestry, health science, etc. In this paper, a bivariate compound Poisson distribution is proposed and the joint probability function of this model is derived. Expressions for the product moments, cumulants, covariance and correlation coefficient are also obtained. Then, an algorithm is prepared in Maple to obtain the probabilities quickly and an empirical comparison of the proposed probability function is given. Bivariate versions of the Neyman type A, Neyman type B, geometric-Poisson, Thomas distributions are introduced and the usefulness of these distributions is illustrated in the analysis of earthquake data.
Key words: Bivariate distribution, Coefficient of correlation, Compound Poisson distribution, Cumulant, Moment.
La distribución compuesta de Poisson univariada tiene muchas aplicaciones en diversas áreas tales como biología, ciencias de la salud, ingeniería forestal, sismología y teoría del riesgo, entre otras. En este artículo, una distribución compuesta de Poisson bivariada es propuesta y la función de probabilidad conjunta de este modelo es derivada. Expresiones para los momentos producto, acumuladas, covarianza y el coeficiente de correlación respectivos son obtenidas. Finalmente, un algoritmo preparado en lenguaje Maple es descrito con el fin de calcular probabilidades asociadas rápidamente y con el fin de hacer una comparación de la función de probabilidad propuesta. Se introducen además versiones bivariadas de las distribuciones tipo A y tipo B de Neyman, geométrica-Poisson y de Thomas y se ilustra la utilidad de estas distribuciones aplicadas al análisis de datos de terremoto.
Palabras clave: coeficiente de correlación, conjuntas, distribución bivariada, distribución compuesta de Poisson, momento.
Texto completo disponible en PDF
References
1. Agnew, D. C. & Jones, L. M. (1991), 'Prediction probabilities from foreshocks', Journal of Geophysical Research 96(11), 959-971. [ Links ]
2. Ambagaspitiya, R. (1998), 'Compound bivariate Lagrangian Poisson distributions', Insurance: Mathematics and Economics 23(1), 21-31. [ Links ]
3. Bruno, M. G., Camerini, E., Manna, A. & Tomassetti, A. (2006), 'A new method for evaluating the distribution of aggregate claims', Applied Mathematics and Computation 176, 488-505. [ Links ]
4. Cacoullos, T. & Papageorgiou, H. (1980), 'On some bivariate probability models applicable to traffic accidents and fatalities', International Statistical Review 48, 345-346. [ Links ]
5. Christophersen, A. & Smith, E. G. C. (2000), A global model for aftershock behaviour, '', Proceedings of the 12th World Conference on Earthquake Engineering, Auckland, New Zealand. [ Links ] Paper 0379.
6. Cossette, H., Gaillardetz, P., Marceau, E. & Rioux, J. (2002), 'On two dependent individual risk models', Insurance: Mathematics and Economics 30, 153-166. [ Links ]
7. Hesselager, O. (1996), 'Recursions for certain bivariate counting distributions and their compound distributions', ASTIN Bulletin 26, 35-52. [ Links ]
8. Holgate, P. (1964), 'Estimation for the bivariate Poisson distribution', Biometrika 51, 241-245. [ Links ]
9. Homer, D. L. (2006), Aggregating bivariate claim severities with numerical fourier inversion, Report , CAS Research Working Party on Correlations and Dependencies among all Risk Sources. 205-230. [ Links ]
10. Johnson, N. L., Kotz, S. & Balakrishnan, N. (1997), Discrete Multivariate Distributions, Wiley, New York. [ Links ]
11. Jones, L. M. (1985), 'Foreshocks and time-dependent earthquake hazard assessment in Southern California', Bulletin of the Seismological Society of America 75, 1669-1679. [ Links ]
12. Kalyoncuoglu, Y. (2007), 'Evaluation of seismicity and seismic hazard parameters in turkey and surrounding area using a new approach to the Gutenberg-Richter relation', Journal of Seismology 11, 31-148. [ Links ]
13. Kocherlakota, S. & Kocherlakota, K. (1992), Bivariate Discrete Distributions, Marcel Decker, New York. [ Links ]
14. Kocherlakota, S. & Kocherlakota, K. (1997), Bivariate discrete distributions, 'Encyclopedia of Statistical Sciences-Update', Vol. 2, Wiley, New York, p. 68-83. [ Links ]
15. Kocyigit, A. & Ozacar, A. (2003), 'Extensional neotectonic regime through the ne edge of the outher isparta angle, SW Turkey: New field and seismic data', Turkish Journal of Earth Sciences 12, 67-90. [ Links ]
16. Özel, G. & Inal, C. (2008), 'The probability function of the compound Poisson process and an application to aftershock sequences', Environmetrics 19, 79-85. [ Links ]
17. Özel, G. & Inal, C. (2010), 'The probability function of a geometric Poisson distribution', Journal of Statistical Computation and Simulation 80, 479-487. [ Links ]
18. Özel, G. & Inal, C. (2011), 'On the probability function of the first exit time for generalized Poisson processes', Pakistan Journal of Statistics 27(4). In press. [ Links ]
19. Panjer, H. (1981), 'Recursive evaluation of a family of compound distributions', ASTIN Bulletin 12, 22-26. [ Links ]
20. Papageorgiou, H. (1997), Multivariate discrete distributions, 'Encyclopedia of Statistical Sciences-Update', Vol. 1, Wiley, New York, p. 408-419. [ Links ]
21. Rolski, T., Schmidli, H., Schmidt, V. & Teugels, J. (1999), Stochastic Processes for Insurance and Finance, John Wiley and Sons. [ Links ]
22. Sastry, N. (1997), 'A nested frailty model for survival data with an application to the study of child survival in Northeast Brazil', Journal of the American Statistical Association 92(438), 426-435. [ Links ]
23. Sundt, B. (1992), 'On some extensions of Panjer's class of counting distributions', ASTIN Bulletin 22, 61-80. [ Links ]
24. Wienke, A., Ripatti, S., Palmgren, J. & Yashin, A. (2010), 'A bivariate survival model with compound Poisson frailty', Statistics in Medicine 29(2), 275-283. [ Links ]
Este artículo se puede citar en LaTeX utilizando la siguiente referencia bibliográfica de BibTeX:
@ARTICLE{RCEv34n3a10,
AUTHOR = {Özel, Gamze}, ]]>
TITLE = {{On Certain Properties of A Class of Bivariate Compound Poisson Distributions and an Application to Earthquake Data}},
JOURNAL = {Revista Colombiana de Estadística},
YEAR = {2011},
volume = {34},
number = {3},
pages = {545-566}
}