A new distribution with decreasing, increasing and upside-down bathtub failure rate

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Abstract

The modeling and analysis of lifetimes is an important aspect of statistical work in a wide variety of scientific and technological fields. For the first time, the so-called generalized exponential geometric distribution is introduced. The new distribution can have a decreasing, increasing and upside-down bathtub failure rate function depending on its parameters. It includes the exponential geometric (Adamidis and Loukas, 1998), the generalized exponential (Gupta and Kundu, 1999) and the extended exponential geometric (Adamidis et al., 2005) distributions as special sub-models. We provide a comprehensive mathematical treatment of the distribution and derive expressions for the moment generating function, characteristic function and rth moment. An expression for Rényi entropy is obtained, and estimation of the stress–strength parameter is discussed. We estimate the parameters by maximum likelihood and obtain the Fisher information matrix. The flexibility of the new model is illustrated in an application to a real data set.

Introduction

When modeling monotone hazard rates, the Weibull distribution may be an initial choice because of its negatively and positively skewed density shapes. However, the Weibull distribution does not provide a reasonable parametric fit for modeling phenomenon with non-monotone failure rates, such as the upside-down bathtub failure rates, which are common in reliability and biological studies. For example, such failure rates curves can be observed in the course of a disease whose mortality reaches a peak after some finite period and then declines gradually. The lifetime models that present upside-down bathtub shaped failure rates are very useful in survival analysis. Recently, probability distributions with such properties have been investigated in Shen et al. (2009), Navarro et al. (2009), Carrasco et al. (2009), Barreto-Souza et al. (2009), Ghitany et al. (2009) and Bebbington et al. (2008).

Adamidis and Loukas (1998) introduced the exponential geometric (EG) distribution whose cumulative distribution function (cdf) is given by FEG(x;β,p)=1eβx1peβx,x>0, for p(0,1) and β>0. In the same way, Kus (2007) introduced the exponential-Poisson distribution. Gupta and Kundu, 1999, Gupta and Kundu, 2001a, Gupta and Kundu, 2001b, Gupta and Kundu, 2007 proposed another lifetime distribution which referred to the generalized exponential (GE) (also called the exponentiated exponential) distribution, and investigated several of its mathematical properties. See, also, Raqab and Ahsanullah (2001). The GE distribution can be defined by elevating the CDF of the exponential distribution with a power α>0.

A further exponentiated type distribution has been introduced and studied in the literature. The exponentiated Weibull (EW) distribution was proposed by Mudholkar and Srivastava (1993) to extend the GE distribution. This distribution was also studied by Mudholkar et al. (1995), Mudholkar and Hutson (1996) and Nassar and Eissa (2003). Nadarajah and Kotz (2006a) introduced four more exponentiated type distributions: the exponentiated gamma, exponentiated Weibull, exponentiated Gumbel and exponentiated Fréchet distributions by generalizing the gamma, Weibull, Gumbel and Fréchet distributions in the same way that the GE distribution extends the exponential distribution. In a recent paper, Barreto-Souza and Cribari-Neto (2009) introduced the generalized exponential-Poisson distribution which extends the exponential-Poisson distribution in the same way that the GE distribution extends the exponential distribution.

Following the same idea of the GE distribution, the CDF of the generalized exponential geometric (GEG) distribution is defined from Eq. (1)F(x;β,p,α)=(1eβx1peβx)α,x>0, where β>0, p(0,1) and α>0.

We now give a characterization for the GEG distribution. We consider α>0 integer and Y1,,Yα a random sample from the EG distribution. For an interpretation of the EG distribution, see Adamidis and Loukas (1998). The definition X=max{Yi}i=1α leads to the GEG distribution which can be used to model the maximum lifetime of a random sample from the EG distribution. Further, it also has a desirable physical interpretation. If there are n components in a parallel system and the lifetimes of the components are independent and identically distributed having the GEG distribution, then the system lifetime also has the GEG distribution.

The rest of the paper is organized as follows. In Section 2, we present the probability density function (PDF) and failure rate function and provide plots of such functions for selected parameter values. In Section 3, we obtain the moment generating and characteristic functions. We also give the moments of the order statistics. The Rényi entropy is derived in Section 4. Maximum likelihood estimation of the parameters and the expected information matrix are discussed in Section 5. Section 6 deals with the estimation of the stress–strength parameter. An application of the GEG model to real data is illustrated in Section 7. Concluding remarks are given in Section 8.

Section snippets

Density function, failure rate and order statistics

The density of the GEG distribution corresponding to the CDF (2) is f(x;β,p,α)=αβ(1p)eβx(1eβx)α1(1peβx)α+1,x>0, where β>0, p(0,1) and α>0. The parameters β and α are scale and shape parameters, respectively. A random variable X with density function (3) is denoted by XGEG(β,p,α).

Clearly, the EG distribution is obtained from (3) when α=1. The GE distribution comes as the limiting distribution (the limit is defined in terms of the convergence in distribution) of the GEG distribution when

Moment generating function

If a random variable Y has the BE(a,b,λ) distribution, Nadarajah and Kotz (2006b) showed that its moment generating function (mgf) and characteristic function (cf) are MBE(t)=E(etY)=B(j+1t/β,α)B(α,j+1),fort<β, and ϕBE(t)=E(eitX)=B(j+1it/β,α)B(α,j+1), respectively, where i=1 is a complex number.

Combining the expansion (5) and these results, we obtain the MGF and CF of the GEG distribution M(t)=(1p)j=0pjB(j+1t/β,α)B(α,j+1)fort<β, and ϕ(t)=(1p)j=0pjB(j+1it/β,α)B(α,j+1), respectively.

Rényi entropy

The entropy of a random variable X is a measure of uncertainty variation. The Rényi entropy is defined by IR(γ)=11γlog{Rfγ(x)dx}, where γ>0 and γ1. Let f() be the GEG density (3). We have fγ(x)=[αβ(1p)]γeβγx(1eβx)γ(α1)(1peβx)γ(α+1).

If we expand (1peβx)γ(α+1) as in Eq. (4), we obtain fγ(x)=j=0cj(β,p,α,γ)fBE(x;γ(α1),γ+j,β), where cj(β,p,α,γ)=[α(1p)]γβγ1pjΓ(γ(α1)+1)Γ(γ(α+1)+j)Γ(γ+j)Γ(γ(α+1))Γ(αγ+j+1)j!. Hence, fγ(x) can be written as an infinite linear combination of BE

Estimation and Fisher information matrix

We examine maximum likelihood estimation and inference for the GEG distribution. Let x1,,xn be a random sample from XGEG(β,p,α) and let θ=(β,p,α)T be the vector of the model parameters. The log-likelihood function for θ reduces to (θ)=nlog{αβ(1p)}βi=1nxi+(α1)i=1nlog(1eβxi)(α+1)i=1nlog(1peβxi).

The score vector U(θ)=(/β,/p,/α)T, has components easily calculated by differentiating (10). We obtain β=nβi=1nxi+(α1)i=1nxieβxi1eβxip(α+1)i=1nxieβxi1peβxi,p=n1p+(

Estimation of the stress–strength parameter

In the context of reliability, the stress–strength model describes the life of a component which has a random strength X subjected to a random stress Y. The component fails at the instant that the stress applied to it exceeds the strength, and the component will function satisfactorily whenever X>Y. Hence, R=Pr(X>Y) is a measure of component reliability. It has many applications, especially in engineering concepts such as structures, deterioration of rocket motors, static fatigue of ceramic

Application

We present an application of the GEG distribution to a real data set. The data set is given by Birnbaum and Saunders (1969) on the fatigue life of 6061-T6 aluminum coupons cut parallel to the direction of rolling and oscillated at 18 cycles per second. The data set consists of 101 observations with maximum stress per cycle 31,000 psi. The data are: 70, 90, 96, 97, 99, 100, 103, 104, 104, 105, 107, 108, 108, 108, 109, 109, 112, 112, 113, 114, 114, 114, 116, 119, 120, 120, 120, 121, 121, 123,

Concluding remarks

The two-parameter exponential geometric (EG) distribution introduced by Adamidis and Loukas (1998) is generalized by introducing an extra parameter, thus creating the generalized EG model (GEG) with a broader class of hazard functions. This is achieved by (the well known technique) raising the cumulative distribution function of the EG to the power of the extra parameter. A detailed study on the probabilistic characteristics of the distribution is presented. The new model includes the

Acknowledgements

We gratefully acknowledge grants from CAPES and CNPq (Brazil). The authors are also grateful to an associate editor and two referees for helpful comments and suggestions.

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