On the Distribution of the Minimum or Maximum of a Random Number of i.i.d. Lifetime Random Variables

Affiliation(s)

Department of Applied Maths & Statistics, ICMC, Universidade de S?o Paulo, S?o Paulo, Brazil.

Department of Statistics, CCET, Universidade Federal de S?o Carlos, S?o Carlos, Brazil.

Department of Applied Maths & Statistics, ICMC, Universidade de S?o Paulo, S?o Paulo, Brazil.

Department of Statistics, CCET, Universidade Federal de S?o Carlos, S?o Carlos, Brazil.

ABSTRACT

Statisticians are usually concerned with the proposition of new distributions. In this paper we point out that a unified and concise derivation procedure of the distribution of the minimum or maximum of a random number N of indepen-dent and identically distributed continuous random variables*Y*_{i},{*i* = 1,2,…,*N*} is obtained if one compounds the probability generating function of *N* with the survival or the distribution func-tion of *Y*_{i}. Expressions are then derived in closed form for the density, hazard and quantile func-tions of the minimum or maximum. The methodology is illustrated with examples of the distributions proposed by Adamidis and Loukas (1998), Kus (2007), Tahmasbi and Rezaei (2008), Barreto-Souza and Cribari-Neto (2009), Cancho, Louzada, and Barriga (2011) and Louzada, Roman and Cancho (2011).

Statisticians are usually concerned with the proposition of new distributions. In this paper we point out that a unified and concise derivation procedure of the distribution of the minimum or maximum of a random number N of indepen-dent and identically distributed continuous random variables

Cite this paper

F. Louzada, E. Bereta and M. Franco, "On the Distribution of the Minimum or Maximum of a Random Number of i.i.d. Lifetime Random Variables,"*Applied Mathematics*, Vol. 3 No. 4, 2012, pp. 350-353. doi: 10.4236/am.2012.34054.

F. Louzada, E. Bereta and M. Franco, "On the Distribution of the Minimum or Maximum of a Random Number of i.i.d. Lifetime Random Variables,"

References

[1] K. Adamidis and S. Loukas, “A Lifetime Distribution with Decreasing Failure Rate,” Statistics and Probability Letters, Vol. 39, No. 1, 1998, pp. 35-42. doi:10.1016/S0167-7152(98)00012-1

[2] K. Adamidis, T. Dimitrakopoulou and S. Loukas, “On an Extension of the Exponential-Geometric Distribution,” Statistics and Probability Letters, Vol. 73, No. 3, 2005, pp. 259-269. doi:10.1016/j.spl.2005.03.013

[3] C. Kus, “A New Lifetime Distribution Distributions,” Computational Statistics and Data analysis, Vol. 11, No. 9, 2007, pp. 4497-4509. doi:10.1016/j.csda.2006.07.017

[4] R. Tahmasbi and S. Rezaei, “A Two-Parameter Life-Time Distribution with Decreasing Failure Rate,” Computational Statistics and Data Analysis, Vol. 52, No. 8, 2008, pp. 3889-3901. doi:10.1016/j.csda.2007.12.002

[5] W. Barreto-Souza and F. Cribari-Neto, “A Generalization of the Exponential-Poisson Distribution,” Statistics and Probability Letters, Vol. 79, No. 24, 2009, pp. 2493-2500. doi:10.1016/j.spl.2009.09.003

[6] V. Cancho, F. Louzada and G. Barriga, “The Poisson-Exponential Lifetime Distribution,” Computational Statistics and Data analysis, Vol. 55, No. , 2011, pp. 677-686. doi:10.1016/j.csda.2010.05.033

[7] F. Louzada, M. Roman and V. Cancho, “The Complementary Exponential Geometric Distribution: Model, Properties, and a Comparison with Its Counterpart,” Computational Statistics and Data analysis, Vol. 55, No. 8, 2011, pp. 2516-2524. doi:10.1016/j.csda.2011.02.018

[8] M. C. Hsiung, T.-H. Tung2, C.-Y. Chang, Y.-C. Chuang, S.-H. S. K-C. Lee, Y.-P. Chou, H. R, C.-M. Huang, W.-H. Y. C-C. Lin and J. W. M. S. Young, “Long-Term Survival and Prognostic Implications of Chinese Type 2 Diabetic Patients with Coronary Artery Disease after Coronary Artery Bypass Grafting,” Applied Mathematics, Vol. 1, No. 3, 2009, pp. 139-145.

[9] G. S. C. Perdona and F. Louzada-Neto, “A General Hazard Model for Lifetime Data in the Presence of Cure Rate,” Journal of Applied Statistics, Vol. 38, No. 7, 2011, pp. 1395-1405. doi:10.1080/02664763.2010.505948

[1] K. Adamidis and S. Loukas, “A Lifetime Distribution with Decreasing Failure Rate,” Statistics and Probability Letters, Vol. 39, No. 1, 1998, pp. 35-42. doi:10.1016/S0167-7152(98)00012-1

[2] K. Adamidis, T. Dimitrakopoulou and S. Loukas, “On an Extension of the Exponential-Geometric Distribution,” Statistics and Probability Letters, Vol. 73, No. 3, 2005, pp. 259-269. doi:10.1016/j.spl.2005.03.013

[3] C. Kus, “A New Lifetime Distribution Distributions,” Computational Statistics and Data analysis, Vol. 11, No. 9, 2007, pp. 4497-4509. doi:10.1016/j.csda.2006.07.017

[4] R. Tahmasbi and S. Rezaei, “A Two-Parameter Life-Time Distribution with Decreasing Failure Rate,” Computational Statistics and Data Analysis, Vol. 52, No. 8, 2008, pp. 3889-3901. doi:10.1016/j.csda.2007.12.002

[5] W. Barreto-Souza and F. Cribari-Neto, “A Generalization of the Exponential-Poisson Distribution,” Statistics and Probability Letters, Vol. 79, No. 24, 2009, pp. 2493-2500. doi:10.1016/j.spl.2009.09.003

[6] V. Cancho, F. Louzada and G. Barriga, “The Poisson-Exponential Lifetime Distribution,” Computational Statistics and Data analysis, Vol. 55, No. , 2011, pp. 677-686. doi:10.1016/j.csda.2010.05.033

[7] F. Louzada, M. Roman and V. Cancho, “The Complementary Exponential Geometric Distribution: Model, Properties, and a Comparison with Its Counterpart,” Computational Statistics and Data analysis, Vol. 55, No. 8, 2011, pp. 2516-2524. doi:10.1016/j.csda.2011.02.018

[8] M. C. Hsiung, T.-H. Tung2, C.-Y. Chang, Y.-C. Chuang, S.-H. S. K-C. Lee, Y.-P. Chou, H. R, C.-M. Huang, W.-H. Y. C-C. Lin and J. W. M. S. Young, “Long-Term Survival and Prognostic Implications of Chinese Type 2 Diabetic Patients with Coronary Artery Disease after Coronary Artery Bypass Grafting,” Applied Mathematics, Vol. 1, No. 3, 2009, pp. 139-145.

[9] G. S. C. Perdona and F. Louzada-Neto, “A General Hazard Model for Lifetime Data in the Presence of Cure Rate,” Journal of Applied Statistics, Vol. 38, No. 7, 2011, pp. 1395-1405. doi:10.1080/02664763.2010.505948