The complementary exponential power lifetime model

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Abstract

In this paper we propose a new lifetime distribution which can handle bathtub-shaped, unimodal, increasing and decreasing hazard rate functions. The model has three parameters and generalizes the exponential power distribution proposed by Smith and Bain (1975) with the inclusion of an additional shape parameter. The maximum likelihood estimation procedure is discussed. A small-scale simulation study examines the performance of the likelihood ratio statistics under small and moderate sized samples. Three real datasets illustrate the methodology.

Introduction

The exponential power (EP) distribution was firstly introduced as a lifetime model by Smith and Bain (1975). This distribution has been discussed by many authors (see, for example, Leemis, 1986, Rajarshi and Rajarshi, 1988, Chen, 1999). In order to avoid confusion, we point out that in the statistical literature, it is possible to find the term exponential power distribution in a context that is not related with survival analysis but within asymmetrical distributions; see, for example, Hazan et al. (2003) and Delicado and Goria (2008).

Let T be a nonnegative random variable denoting the lifetime of an individual in some population. The random variable T is said to be exponentially powered distributed with scale parameter α>0 and shape parameter β>0 if its probability density function (pdf) is given by fSB(t)=βα(tα)β1exp{(tα)β}exp(1exp{(tα)β}),t>0.

The corresponding survival and hazard functions are given by SSB(t)=exp(1exp{(tα)β}),t>0 and hSB(t)=βα(tα)β1exp{(tα)β},t>0. The shape of hSB(t) depends on the value of the shape parameter β. For β1, the hazard function is increasing. For β<1 the hazard function is bathtub shaped.

In recent years, new classes of models have been proposed based on modifications of the EP model. Chen (2000) proposed a new model with two shape parameters. His model is appealing since though having only two parameters it can accommodate increasing and bathtub shaped hazard functions. Also, it holds some nice properties on the classical inferential front. The confidence intervals for the shape parameters and their joint confidence regions have closed form. However, it lacks a scale parameter that makes it less flexible for analyzing a variety of datasets. To overcome such a limitation, Xie et al. (2002) proposed a model, known as the Weibull extension model, which can be considered as an extension of Chen’s model, with an additional scale parameter. As a result, the model becomes more flexible and persuasive from the point of view of practitioners. The Xie model version however accommodates only increasing and bathtub shaped hazard functions as its antecessors.

Although the EP distribution and its modifications are commonly used for analyzing lifetime data, they do not provide a reasonable parametric fit for some practical applications where the underlying hazard functions may be decreasing or unimodal shaped. For instance, Efron (1988) studied the survival times of patients from a head and neck cancer clinical trial, which have a unimodal shaped hazard function.

In this paper, we propose a new three-parameter distribution family which can accommodate increasing, decreasing, unimodal and bathtub shaped hazard functions. The distribution is mainly related to the distribution proposed by Smith and Bain (1975) with an additional shape parameter, then hereafter it shall be called the Complementary Exponential Power family distribution, or simply the CEP distribution.

In the survival literature there are several flexible distributions which can accommodate increasing, decreasing, unimodal and bathtub shaped hazard functions (see, for example, Mudholkar and Srivastava, 1993, Mudholkar et al., 1996, Pham and Lai, 2007, Carrasco et al., 2008). Some of them are four-parameter distributions, while the CEP is a three-parameter one. This is an advantage from a practical point of view, since it is important to consider parsimonious models with as few parameters as possible. Particularly, with small and moderate sized samples, a usual situation in survival analysis, where the parameters may not be accurately estimated (Xie et al., 2002).

The remainder of this paper is organized as follows. In Section 2 we present the CEP distribution with some special cases. Also we provide a general expression for the kth ordinary moment and discuss the hazard function shape. In Section 3 the inferential procedure based on maximum likelihood is presented. In Section 4 we present the results of a simulation study which evaluate the performance of the LR statistic in the presence of small and moderate samples. In Section 5 three real datasets illustrate the application of the propose CEP distribution and the comparison of the CEP distribution with the modified Weibull distribution proposed by Pham and Lai (2007) and with the four parameters generalized modified Weibull distribution proposed by Carrasco et al. (2008). Concluding remarks in Section 6 finish the paper.

Section snippets

The CEP distribution

As described by Marshall and Olkin (2007), an exponentiated distribution can be easily constructed. It is based on the observation that by raising any baseline cumulative distribution function (cdf) Fbaseline(t) to an arbitrary power θ>0, a new cdf F(t)=Fbaseline(t)θ>0 is obtained, but now with the additional parameter θ, which can be refereed as a resilience parameter and F(t) is a resilience parameter family. Although it is not our case, the term resilience easily emerges if we let θ be an

Inference

We assume that the lifetime are independently distributed, and also are independent from the censoring mechanism. We observe ti=min(Ti0,Ci), where Ti0 is the lifetime for the ith individual with distribution given by (4) and Ci is the censoring time for the ith individual, i=1,,n. In this case the log-likelihood function of α,β and θ is given by (α,β,θ)=rlog(θβ)rβlog(α)+(β1)iFlog(ti)+r+iF(tiα)βiFexp((tiα)β)+(θ1)iFlog(1gα,β(ti))+iClog(1(1gα,β(ti))θ) where F denotes the set of

A simulation study

In this section a Monte Carlo simulation study was carried out with different sample sizes (n=50,100 and 200) in order to evaluate the performance of the LRS at a 5% nominal significance level. The lifetimes denoted by T1,,Tn were generated from the CEP distribution given in (4) considering values for the parameters α=50,β=0.2 and θ=0.25,0.5,1,2,4,8. The censoring times Ci,i=1,,n were generated from a uniform distribution U(0,τ), with τ controlling the percentage of censoring observations.

Applications

In this section we reanalyze three datasets. The first example presents a unimodal hazard function, the second presents a bathtub-shaped hazard function and the third one an increasing hazard function.

In order to identify the shape of a lifetime data hazard function many approaches have been proposed (see, Glaser, 1980). In this context, a graphical method based on the total time on test (TTT) transformed, introduced by Barlow and Campo (1975), shall be considered here. It has been shown that

Concluding remarks

In this paper a new model for lifetime data is provided and discussed. The new CEP distribution generalizes the EP distribution. It is flexible and can accommodate increasing, decreasing, unimodal and bathtub hazard functions. Maximum likelihood inference is implemented straightforwardly and parametric simulation can be used for generating confidence intervals for the parameters and hypothesis testing. The practical importance of the new distribution was demonstrated in three applications,

Acknowledgements

The authors thank the associate editor and the referees for helpful comments which led to several improvements in the text. Francisco Louzada-Neto and Vicente G. Cancho’s researches are granted by CNPq-Brasil.

References (30)

  • A. Davison et al.

    Bootstrap Methods and their Application

    (1997)
  • B. Efron

    Logistic regression, survival analysis, and the Kaplan–Meier curve

    Journal of the American Statistical Association

    (1988)
  • R.E. Glaser

    Bathtub and related failure rate characterizations

    Journal of the American Statistical Association

    (1980)
  • R.D. Gupta et al.

    Generalized exponential distributions

    Australian & New Zealand Journal of Statistics

    (1999)
  • C.D. Lai et al.

    A modified Weibull distribution

    IEEE Transactions on Reliability

    (2003)
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