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 AM  Vol.3 No.4 , April 2012
On the Distribution of the Minimum or Maximum of a Random Number of i.i.d. Lifetime Random Variables
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 Yi,{i = 1,2,…,N} is obtained if one compounds the probability generating function of N with the survival or the distribution func-tion of Yi. 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).
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.
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

 
 
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