On the Distribution of the Minimum or Maximum of a Random Number of i.i.d. Lifetime Random Variables ()
1. Introduction
Several authors have proposed new distributions for the maximum or the minimum as extensions of the exponential distribution, such as [1-7]. 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
, 
, 
being also a strictly positive integer random variable with discrete probability function (dpf) and probability generating function (pgf) 
defined throughout the interval
.
Let 
and 
be the survival function and cumulative distribution function of the random variables
,
. The cumulative distribution of the maximum out of 
is obtained by composing 
with the cumulative distribution function of 
and the survival function of the minimum is obtained by composing 
with the survival function of
.
2. Model Formulation
Let 
be a strictly positive random variable with dpf and pgf of 
defined throughout the interval
. 
is increasing in 
and satisfies the equalities 
and
. Thus 
can be viewed as the value for 
of a cumulative distribution function.
If 
is an absolutely continuous function, its pdf is denoted by
, and risk function is represented by
, both supported on
. Keeping the assumptions made in this section on the random variable
, it follows that the function 
is decreasing function of 
being thus the equalities 
and
. So 
is the restriction to the interval 
of a survival function and if it is absolutely continuous function it has pdf represented by 
and hazard function denoted
, both with support in
.
In this paper 
is the vector of parameters of 
and all other Greek letters refer to the parameters of cumulative distribution function of
, which is represented by
.
Let 
be a sequence of i.i.d. random variables with pdf and 
corresponding to the number of random variables 
i.i.d. random variables with survival function 
and the cumulative distribution function
.
For 
the survival function of 
is given by
, but when 
is a random variable the survival function of the minimum is given by
(1)
Several authors have obtained density functions of the minimum by (2), which requires the calculation of a series, given by
(2)
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,
(3)
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
(4)
The pdf, hazard and quantile functions of the minimum or maximum of the 
are defined respectively as
(5)
(6)
and
(7)
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
.
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.
3. 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), assuming 
as the survival function from an exponentiated random variable. However, many new distributions may be obtained by considering a composition of different 
and 
functions. For instance, assuming
, 
(the geometric pgf) and 
(the Weibull survival function), we obtain 
and
.
Table 1. The pgf of N and survival function and density function (p.d.f) for 
or
.
We fit the five different distributions presented in Table 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]. Serum-reversal can occur in children born from mothers infected with HIV. In order to compare the distributions we consider the 
values, the Akaike information criterion (AIC) and Bayesian information criterion (BIC). The best distribution corresponds to lower
, AIC and BIC values. The Table 2 shows the parameter MLEs and their corresponding standard errors in 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 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].
Table 2. The parameter MLEs, their corresponding standard errors in parentheses, values of the –LOG, AIC and BIC to the five fitted distributions.
Figure 1. Fitted density functions on the histogram (left panel), and survival function (right panel) of the EP, EG, CEG,PE and El distributions superimposed to the Kaplan-Meier fit.
4. Acknowledgements
The authors thank the referees for their comments. The research of Francisco Louzada is supported by the Brazilian organization CNPq.