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

Francisco Louzada, Estela M. P. Bereta, Maria A. P. Franco

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.

DOI: 10.4236/am.2012.34054                

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).

Keywords

Compounding Distributions; Distribution of the Maximum; Distribution of the Minimum; Probability Generating Function

Conflicts of Interest

The authors declare no conflicts of interest.

References

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