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

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 independent and identically distributed continuous random variables , i Y   = 1, 2, , i  N is obtained if one compounds the probability generating function of N with the survival or the distribution function of Y i. Expressions are then derived in closed form for the density, hazard and quantile functions of the minimum or maximum. The methodology is illustrated with examples of the distributions proposed by


Introduction
Several authors have proposed new distributions for the maximum or the minimum as extensions of the exponential distribution, such as [1][2][3][4][5][6][7].In this paper, we obtain an alternative form to the one considered by these authors for obtaining the distribution of the minimum or maximum of independent and identically distributed (i.i.d.) random variables i , .Keeping the assumptions made in this section on the random variable , it follows that the function  , but when is a random variable the survival function of the minimum is Several authors have obtained density functions of the minimum by (2), which requires the calculation of a series, given by ( In this paper we show that a more concise way to obtain the functions that determine the distribution of the minimum without the need of the calculation a series by considering the fact that the expression (1) can also be written as, Thus, the survival function of the minimum is obtained directly from (3), consequently the pdf of the minimum is obtained by derivation of .Similarly, the survival function of maximum is obtained from , and the cumulative distribution and pdfs of the maximum are obtained by derivation of .From (3) follows that the survival function of the minimum and the cumulative distribution function of are defined as The pdf, hazard and quantile functions of the mini- are defined respectively as , . and where is the quantile function the of basic distribution of .
The maximum likelihood estimates (MLEs) of the parameters are obtained by direct maximization of the loglikelihood function, , or ma .
The advantage of this procedure is that it runs immediately using existing statistical packages such as R. The EM-algorithm can also be considered as in [6].Largesample inference for the parameters can be based on their MLEs and estimated standard errors, or, preferably, on the profile likelihood, the later being invariant under reparametrization and a safer guide in relatively small samples.Different approaches are via the bootstrap or via Bayesian inference.

Some Working Examples
Table 1 shows the pgf of , the survival function and the density function of the minimum or maximum of i.i.d.random variables for the distributions proposed by [1,3,4,6,7], obtained respectively by considering ( 4), ( 5) and ( 6 we obtain

  =1
= min  1 presents the fitted density functions on the histogram, and survival function of the EG, EP, EL, PE and CEG distributions superimposed to the data histogram and Kaplan-Meier fit, respectively.The presence of long-term survivals is very common in practice [8].Our approach should be investigate in the long-term survival context.A possible approach is to consider the mixture model adopted by [9].
being also a strictly positive integer random variable with discrete probability function (dpf) and probability generating function (pgf) function and cumulative distribution function of the random variables i , .The cumulative distribution of the maximum out of Y with the cumulative distribution function of i and the survival function of the minimum is obtained by composing Y Let be a strictly positive random variable with dpf and pgf of N function from an exponentiated random variable.However, many new distributions may be obtained by considering a composition of different parentheses, values of the , AIC and BIC.The values of AIC, BIC and provide evidence in favor the CEG distribution.These results are corroborated by the fitted density functions and survival functions of the five distributions superimposed to the histogram and Kaplan-Meier curve.The Figure

Figure 1 .
Figure 1.Fitted density functions on the histogram (left panel), and survival function (right panel) of the EP, EG, CEG,PE nd El distributions superimposed to the Kaplan-Meier fit. a

Table 1 . The pgf of N and survival function and density function
(p.d.f) for or .

Table 2 . The parameter MLEs, their corresponding standard errors in parentheses, values of the -LOG, AIC and BIC to the five fitted distributions.
[8]fit the five different distributions presented in Ta- ble 1 in a real data set on the serum-reversal time (days) of 143 children contaminated with HIV by vertical transmission from the University Hospital of the Ribeiro Preto School of Medicine (Hospital das Clnicas da Faculdade de Medicina de Ribeiro Preto) from 1986 to 2001[8].