﻿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) , Article ID: 18884 , 4 pages DOI:10.4236/am.2012.34054

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

Francisco Louzada1, Estela M. P. Bereta2, Maria A. P. Franco2

1Department of Applied Mathematics & Statistics, ICMC, Universidade de São Paulo, São Paulo, Brazil

2Department of Statistics, CCET, Universidade Federal de São Carlos, São Carlos, Brazil

Received January 19, 2012; revised March 5, 2012; accepted March 12, 2012

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

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 independent and identically distributed continuous random variables , is obtained if one compounds the probability generating function of N with the survival or the distribution function of Yi. 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 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).

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 . 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 . 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 . Our approach should be investigate in the long-term survival context. A possible approach is to consider the mixture model adopted by . 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.

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