www.sjmmf.org Journal of Modern Mathematics Frontier Volume 3 Issue 1, March 2014 doi: 10.14355/jmmf.2014.0301.03 A Note on Exponentiated -geometric Distributions Hamid Bidram, S. Marzieh Alavi Department of Statistics, University of Isfahan, Isfahan, 81746-73441, Iran h.bidram@sci.ui.ac.ir, s.marziehalavi@yahoo.com Abstract Let 1 , 2 ,… be a random sample from an absolutely continuous distribution , where has a geometric distribution and is independent of ’. In recent years, several compound distributions, called F-geometric distributions, have been generated by the smallest or largest order statistic of ’ in the literature. Further, exponentiated F-geometric (EFG) distributions, belonging to the resilience parameter family, have been introduced by some authors recently. In this paper, using Rohatgi’s (1987) results, we firstly obtain a simple closed form for the cumulative distribution function of the rth order statistic of ’ and reveal a relationship between it and EFG distributions. Finally, a new EFG distribution with its application in data modeling is introduced. Keywords Beta Generalized Exponential Distribution; Compounding; Order Statistics; Random Sample Size; Resilience Parameter Family Introduction For many years, authors have been interested in developing methods for generating distributions with higher flexibility in applications and data modeling. Compounding mechanism is one of the methods for generalizing distributions, which has received much attention in the literature. Let 1 , 2 ,… be independent and identically distributed (iid) random variables from an absolutely continuous distribution , where is itself a random variable with a geometric distribution, independent of ’. Then, several new distributions can be generated by the smallest ( = {} =1 ) or largest ( = max{ } =1 ) order statistic which are ordinary called -geometric distributions. Some remarkable distributions in this connection can be addressed as exponential-geometric (EG) distribution of Adamidis and Loukas (1998), a complementary version of the EG distribution proposed by Adamidis et al. (2005) and also Louzada et al. (2011), Weibull-geometric (WG) distribution of Barreto-Souza et al. (2011) and its complementary version given by Tojeiro et al. (2014), new generalized exponential geometric (NGEG) distribution of Bidram et al. (2013a), and exponentiated exponential- geometric (E2G) distribution of Louzada et al. (2014). An alternative method of generalizing distributions is to elevate the cumulative distribution function (cdf), say , of a random variable with a power parameter, say >0 . The parameter >0 is called a resilience parameter and {(. |)=[(. )] , > 0} is a resilience parameter family or, alternatively, an exponentiated F distribution (see Marshall and Olkin, 2007). Generalized exponential (GE) distribution of Gupta and Kundu (1999) and exponentiated Weibull (EW) distribution of Mudholkar and Hutson (1996) are two important distributions belonging to the resilience parameter family. Recently, some new distributions have been proposed by elevating the cdf of an -geometric distribution with the power of a resilience parameter, called exponentiated -geometric (EFG) distribution. For a random variable, say , with an EFG distribution, we write ~(, , ), where θ is parameter vector of F distribution, 0 < < 1 is the parameter of the geometric distribution, and > 0 is the resilience parameter. Two important distributions belonging to the EFG family are generalized exponential geometric (GEG) (or exponentiated exponential geometric (EEG)) distribution of Silva et al. (2010) and exponentiated Weibull-geometric (EWG) distribution which is a special case of beta Weibull-geometric (BWG) distribution of Cordeiro et al. (2011) and Bidram et al. (2013b). In recent years, compound distributions have often been generated by = { } =1 or = max{ } =1 . In this paper, it is shown that the distribution of the rth order statistic, as an extension of the distribution of U, is easily obtained using a corresponding EFG distribution. Indeed, it is revealed that the th order statistic of a random sample of size N drawn from an absolutely continuous distribution, say F, is identically distributed with the random variable ~(, , ) when > 0 is an integer value. Main Results In this section, the main results of this paper are given 18