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DISTRIBUTION FUNCTION OF FIRST EXIT TIME FOR A COMPOUND POISSON PROCESS

Year 2011, Volume: 1 Issue: 2, 91 - 104, 04.07.2011

Gamze Özel Ceyhan İnal

Abstract

In this study, the first exit time of a compound Poisson process with positive jumps and an upper horizontal boundary is considered. An explicit formula is derived for the distribution function of the first exit time associated with the compound Poisson process. By means of a proposed algorithm, some numerical examples and an application on traffic accidents are also given to illustrate the usage of the distribution function of the first exit time and proposed algorithm.

Keywords

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References

  • Ammussen, S. (2000). Ruin Probabilities. World Scientific.
  • Bar-Lev, S., Bshouty, D., Perry, D. and Zacks, S. (1999). Distributions of stopping times for com- pound Poisson processes. Stochastic Models 15(1), 89-101.
  • Cacoullos, T. and Papageorgiou, H. (1980). On some bivariate probability models applicable to traffic accidents and fatalities. International Statistical Review 48, 345-356.
  • Cacoullos, T. and Papageorgiou, H. (1982). Bivariate negative binomial-Poisson and negative bino- mial-Bernoulli models with an application to accident data. In Statistics and Probability: Essays in Honor of C. R. Rao, G. Kallianpur et al. (eds) 155-168.
  • Delucia, J. and Poor, H.V. (1997). The moment generating function of the stopping time for a linearly stopped Poisson process. Stochastic Models 13, 275-292.
  • Dirickx, Y.M.I. and Koevoets, D. (1997). A continuous review inventory model with compound Pois- son demand process and stochastic lead time. Naval Research Logistics Quarterly 24(4), 577.
  • Djauhari, M.A. (2002). Stochastic pattern of traffic accidents in Bandung city, IATSS Research 26(2), 85-91.
  • Dvoretzky, A., Kiefer, J. and Wolfowitz, J. (1953). Sequential decision problems for processes with continuous time parameter. Annals of Mathematical Statistics 24, 254-264.
  • Gallot, S.F.L. (1983). Absorption and first passage times for a compound process in a general upper boundary. Journal of Applied Probability 30, 835-850.
  • Ignatov, Z.G. and Kaishev, V.K. (2004). A finite-time ruin probability formula for continuous claim severites. Journal of Applied Probability 41(2), 570-578.
  • Klugman, S.A., Panjer, H.H. and Willmot, G.E. (1998). Loss Models. John Wiley & Sons.
  • Leiter, R.E. and Hamdan, H.A. (1973). Some bivariate probability models applicable to traffic acci- dents and fatalities. International Statistical Review 41, 87-100.
  • Meintanis, S.G. (2007). A new goodness of fit test for certain bivariate distributions applicable to traf- fic accidents. Statistical Methodology 4, 22-34.
  • Ozel, G. and Inal, C. (2008). The probability function of the compound Poisson process and an appli- cation to aftershock sequence in Turkey. Environmetrics 19(1), 79-85.
  • Ozel, G. and Inal, C. (2008). The probability function of the compound Poisson distribution using in- teger partitions and Ferrer's graph. Bulletin of Statistics and Economics 2, 70-79, (2008).
  • Panjer, H.H. (1981). Recursive evaluation of a family of compound distributions. Astin Bulletin 12, 22-26.
  • Rolski, T., Schmidli, H., Schmidt, V. and Teugels, J. (1998). Stochastic Processes for Insurance and Finance. John Wiley & Sons.
  • Rosencrantz, W.A. (1983). Calculation of the Laplace transform of the length of the busy period for the M/G/1 queue via martingales. Annals of Probability 11, 817-818.
  • Shanmungan, R. and Singh, J. (1980). Some bivarite probability models applicable to traffic accidents and fatalities. Proceedings of the NATO Advanced Study Institute: Statistical Distributions in Scientific Work, 6, 95-103.
  • Van Der Laan, B.S. and Louter, A.S. (1986). A statistical model for the costs of passenger car traffic accidents. The Statistician 35(2), 163-174.

BİRLEŞİK POISSON SÜRECİ İÇİN İLK AŞMA ZAMANININ DAĞILIM FONKSİYONU

Year 2011, Volume: 1 Issue: 2, 91 - 104, 04.07.2011

Gamze Özel Ceyhan İnal

Abstract

Bu çalışmada, pozitif adımlı ve üst yatay sınırlı bir birleşik Poisson sürecinin ilk aşma zamanı göz önünde bulundurulmuş ve ilk aşma zamanının kesin formülüne ulaşılmıştır. İlk aşma zamanının dağılım fonksiyonunun ve önerilen algoritmanın uygulanabilirliğini göstermek için bazı sayısal örnekler ve trafik kazaları üzerine bir uygulama verilmiştir

Keywords

Birleşik poisson süreci İlk aşma zamanı Üst yatay sınır Oracle veritabanı

References

  • Ammussen, S. (2000). Ruin Probabilities. World Scientific.
  • Bar-Lev, S., Bshouty, D., Perry, D. and Zacks, S. (1999). Distributions of stopping times for com- pound Poisson processes. Stochastic Models 15(1), 89-101.
  • Cacoullos, T. and Papageorgiou, H. (1980). On some bivariate probability models applicable to traffic accidents and fatalities. International Statistical Review 48, 345-356.
  • Cacoullos, T. and Papageorgiou, H. (1982). Bivariate negative binomial-Poisson and negative bino- mial-Bernoulli models with an application to accident data. In Statistics and Probability: Essays in Honor of C. R. Rao, G. Kallianpur et al. (eds) 155-168.
  • Delucia, J. and Poor, H.V. (1997). The moment generating function of the stopping time for a linearly stopped Poisson process. Stochastic Models 13, 275-292.
  • Dirickx, Y.M.I. and Koevoets, D. (1997). A continuous review inventory model with compound Pois- son demand process and stochastic lead time. Naval Research Logistics Quarterly 24(4), 577.
  • Djauhari, M.A. (2002). Stochastic pattern of traffic accidents in Bandung city, IATSS Research 26(2), 85-91.
  • Dvoretzky, A., Kiefer, J. and Wolfowitz, J. (1953). Sequential decision problems for processes with continuous time parameter. Annals of Mathematical Statistics 24, 254-264.
  • Gallot, S.F.L. (1983). Absorption and first passage times for a compound process in a general upper boundary. Journal of Applied Probability 30, 835-850.
  • Ignatov, Z.G. and Kaishev, V.K. (2004). A finite-time ruin probability formula for continuous claim severites. Journal of Applied Probability 41(2), 570-578.
  • Klugman, S.A., Panjer, H.H. and Willmot, G.E. (1998). Loss Models. John Wiley & Sons.
  • Leiter, R.E. and Hamdan, H.A. (1973). Some bivariate probability models applicable to traffic acci- dents and fatalities. International Statistical Review 41, 87-100.
  • Meintanis, S.G. (2007). A new goodness of fit test for certain bivariate distributions applicable to traf- fic accidents. Statistical Methodology 4, 22-34.
  • Ozel, G. and Inal, C. (2008). The probability function of the compound Poisson process and an appli- cation to aftershock sequence in Turkey. Environmetrics 19(1), 79-85.
  • Ozel, G. and Inal, C. (2008). The probability function of the compound Poisson distribution using in- teger partitions and Ferrer's graph. Bulletin of Statistics and Economics 2, 70-79, (2008).
  • Panjer, H.H. (1981). Recursive evaluation of a family of compound distributions. Astin Bulletin 12, 22-26.
  • Rolski, T., Schmidli, H., Schmidt, V. and Teugels, J. (1998). Stochastic Processes for Insurance and Finance. John Wiley & Sons.
  • Rosencrantz, W.A. (1983). Calculation of the Laplace transform of the length of the busy period for the M/G/1 queue via martingales. Annals of Probability 11, 817-818.
  • Shanmungan, R. and Singh, J. (1980). Some bivarite probability models applicable to traffic accidents and fatalities. Proceedings of the NATO Advanced Study Institute: Statistical Distributions in Scientific Work, 6, 95-103.
  • Van Der Laan, B.S. and Louter, A.S. (1986). A statistical model for the costs of passenger car traffic accidents. The Statistician 35(2), 163-174.
There are 20 citations in total.

Details

Primary Language English
Journal Section Articles
Authors

Gamze Özel

Ceyhan İnal This is me

Publication Date July 4, 2011
Published in Issue Year 2011 Volume: 1 Issue: 2

Cite

APA Özel, G., & İnal, C. (2011). DISTRIBUTION FUNCTION OF FIRST EXIT TIME FOR A COMPOUND POISSON PROCESS. Anadolu University Journal of Science and Technology B - Theoretical Sciences, 1(2), 91-104.
AMA Özel G, İnal C. DISTRIBUTION FUNCTION OF FIRST EXIT TIME FOR A COMPOUND POISSON PROCESS. AUBTD-B. August 2011;1(2):91-104.
Chicago Özel, Gamze, and Ceyhan İnal. “DISTRIBUTION FUNCTION OF FIRST EXIT TIME FOR A COMPOUND POISSON PROCESS”. Anadolu University Journal of Science and Technology B - Theoretical Sciences 1, no. 2 (August 2011): 91-104.
EndNote Özel G, İnal C (August 1, 2011) DISTRIBUTION FUNCTION OF FIRST EXIT TIME FOR A COMPOUND POISSON PROCESS. Anadolu University Journal of Science and Technology B - Theoretical Sciences 1 2 91–104.
IEEE G. Özel and C. İnal, “DISTRIBUTION FUNCTION OF FIRST EXIT TIME FOR A COMPOUND POISSON PROCESS”, AUBTD-B, vol. 1, no. 2, pp. 91–104, 2011.
ISNAD Özel, Gamze - İnal, Ceyhan. “DISTRIBUTION FUNCTION OF FIRST EXIT TIME FOR A COMPOUND POISSON PROCESS”. Anadolu University Journal of Science and Technology B - Theoretical Sciences 1/2 (August 2011), 91-104.
JAMA Özel G, İnal C. DISTRIBUTION FUNCTION OF FIRST EXIT TIME FOR A COMPOUND POISSON PROCESS. AUBTD-B. 2011;1:91–104.
MLA Özel, Gamze and Ceyhan İnal. “DISTRIBUTION FUNCTION OF FIRST EXIT TIME FOR A COMPOUND POISSON PROCESS”. Anadolu University Journal of Science and Technology B - Theoretical Sciences, vol. 1, no. 2, 2011, pp. 91-104.
Vancouver Özel G, İnal C. DISTRIBUTION FUNCTION OF FIRST EXIT TIME FOR A COMPOUND POISSON PROCESS. AUBTD-B. 2011;1(2):91-104.
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