A New Five-Parameter Lifetime Model: Theory and Applications

In this paper we defined a new lifetime model called the the Exponentiated additive Weibull (EAW) distribution. The proposed distribution has a number of well-known lifetime distributions as special submodels, such as the additive Weibull, exponentiated modified Weibull, exponentiated Weibull and generalized linear failure rate distributions among others. We obtain quantile, moments, moment generating functions, incomplete moment, residual life and reversed Failure Rate Functions, mean deviations, Bonferroni and Lorenz curves. The method of maximum likelihood is used for estimating the model parameters. Applications illustrate the potentiality of the proposed distribution.


Introduction
In reliability engineering and lifetime analysis many applications require a bathtub shaped hazard rate function. Weibull distribution is one of the most commonly used lifetime distributions in reliability and lifetime data analysis. It is flexible in modeling failure time data, as the corresponding hazard rate function can be increasing,constant or decreasing. But in many applications in reliability, mechanical and electronic components and survival analysis, the hazard rate function can be of bathtub shape. It is well known that, because of design and manufacturing problems, the hazard rate is high at the beginning of a product life cycle and decreases toward a constant level. After reaching a certain age, the product enters wear-out phase and hazard rate starts to increase. Despite the fact that this phenomenon has been presented in many reliability engineering texts few practical models possessing this property have appeared in the literatures twoparameter flexible Weibull extension of Bebbington et al. (2007) has a hazard function that can be increasing, decreasing or bathtub shaped. Zhang and Xie (2011) studied the characteristics and application of the truncated Weibull distribution which has a bathtub shaped hazard function. A three parameter model, called exponentiated Weibull distribution, was introduced by Mudholkar and Srivastave (1993). Xie and Lai (1995) introduced a four-parameter distribution called the additive Weibull distribution based on the simple idea of combining the hazard rates of two Weibull distributions: one has a decreasing hazard rate and the other one has an increasing hazard rate. It has the cumulative distribution function is given by ( , , , , ) = 1 − −( + ) ; > 0, (1) where > 0, > 0 and > > 0, or > > 0 which gives identifiability to the model, when > 0 the hazard rate is increasing and when 0 < < 1 hazard rate is decreasing. The corresponding probability density function is where > 0 and > 0 are scale parameters, and > > 0, or ( > > 0) are shape parameters. The interpretation of model (2) is evident. Suppose a system composed of two interconnected independent series sub-systems that affect the system in a different way, each one having a Weibull distribution with proper parameters. The hazard time of the system follows (2), since it occurs when the first of the two sub-systems fails.
Since 1995, exponentiated distributions have been widely studied in statistics and numerous authors have developed various classes of these distributions. A good review of some of these models is presented by Pham and Lai (2007). The exponentiation of distributions is a mechanism that makes the model more flexible, Nadarajah  In this paper, the so-called exponentiated additive Weibull (EAW) distribution with five parameters is proposed. The new distribution due to its flexibility in accommodating all the forms of the hazard rate function can be used in a variety of problems for modeling lifetime data. Another important characteristic of the distribution is that it contains, as special sub-models, the Weibull, exponentiated exponential Kundu, 1999, 2001), exponentiated modified Weibull distribution (Elbatal, 2009), exponentiated Weibull distribution (Mudholkar et al., 1995(Mudholkar et al., , 1996, generalized linear failure rate distribution, (Sarhan and Kundu, 2009), Modified Weibull distribution (Lai et al, 2003), among some other distributions. The exponentiated additive Weibull distribution is not only convenient for modeling comfortable bathtub-shaped failure rates data but is also suitable for testing goodness-of-fit of some special sub-models such as the exponentiated Weibull and modified Weibull distributions.
The rest of the article can be organized as follows. In section 2 we present the expression of the pdf and cdf of the subject distribution and some special sub-models. In section 3 we study the statistical properties including moments, moment generating function and incomplete moments. Residual life and reversed residual functions of the distribution, Bonferroni and Lorenz curves and mean deviations are discussed in Section 4. In section 5 we demonstrate the maximum likelihood estimates of the unknown parameters. Simulation results to assess the performance of the maximum likelihood estimation method are reported in Section 6. Finally, in section 7 we present a data analysis to illustrate the usefulness of the proposed distribution.

Exponentiated Additive Weibull Distribution
A random variable has the distribution with parameter vector = ( , , , , ) say, ( ), or ( , , , , ) if its cumulative distribution given by and its probability density function given by The survival function, also known as the reliability function (rf) in engineering, is the characteristic of an explanatory variable that maps a set of events, usually associated with mortality or failure of some system onto time. The corresponding survival function of random variable is and the hazard rate (failure) function (hrf) which is an important quantity characterizing life phenomenon functions takes the following form ; > 0 (6) whereas its reversed hazard rate (failure) function (rhf) is given by   It is known, not many lifetime distributions exhibit bathtub hazard rates. The model shows flexibility in accommodating all forms of the hazard rate function as seen from Figure 2 (by changing its parameter values) seems to be an important distribution that can be used.

Special Cases of the Distribution
The exponentiated additive Weibull distribution is a very flexible model that approaches to different distributions when its parameters vary. The flexibility of the exponentiated additive Weibull distribution is explained in

Statistical Properties
In this section we studied the shapes, statistical properties, specifically moments, incomplete moment, and moment generating function of the ( ) distribution.

Moments
In this section, the different moments of the exponentiated additive Weibull distribution can be obtained using the ℎ moment ′ = ( ) and the moment generating function, ( ) = .
Theorem 1 The ℎ moment of distribution, = 1,2, . ... is given by Proof We start with the well known definition of the ℎ moment of the random variable with probability density function ( ) given by Substituting from (4) into the above relation, we get since 0 < −( + ) < 1 for > 0, then by using the binomial series expansion of is given by we get (11) but the series expansion of −( +1) is given by substituting from (12) into (11), we get where , after some algebra, the integral in (13) can be computed as follows which completes the proof.
The central moments and cumulants of the distribution can be determined from expression (8) 4 , etc. Additionally, the skewness and kurtosis can be calculated from the third and fourth standardized cumulants in the forms = 3 /√ 2 3 and = 4 / 2 2 , respectively.

Theorem 2
The moment generating function of distribution is given by Proof We start with the well known definition of the moment generating function given by which completes the proof.
Similarly, the characteristic function of the distribution becomes ( ) = ( ) where = √−1 is the unit imaginary number.

Conditional Moments
The main application of the first incomplete moment refers to the Bonferroni and Lorenz curves. These curves are very useful in economics, reliability, demography, insurance and medicine. The answers to many important questions in economics require more than just knowing the mean of the distribution, but its shape as well. This is obvious not only in the study of econometrics but in other areas as well. For lifetime models, it is also of interest to find the conditional moments and the mean residual lifetime function. The conditional moments for distribution is given by ]. (16) where is the upper incomplete gamma function. The mean residual lifetime function is given by

Quantile Function
The quantile function (qf) of is obtained by inverting (3), but there is no colsed form for the qf of distribtion. The ℎ quantail of the distribtion can be obtained by numerically solving the following equaiton for One can simulate the random variable with = ( , , , , ) by: 1. Generate ~(0,1). (18). 3. Solve equation (18) numerically for .

Put = in equation
The median can be calculated by putting = . in equation (18), and solving it numerically for .

Residual life and Reversed Failure Rate Function
Given that a component survives up to time ≥ 0, the residual life is the period beyond t until the time of failure and defined by the conditional random variable − | > . In reliability, it is well known that the mean residual life function and ratio of two consecutive moments of residual life determine the distribution uniquely (Gupta and Gupta, 1983). Therefore, we obtain the ℎ order moment of the residual lifetime can be obtained via the general formula Applying the binomial expansion of ( − ) into the above formula, we get The mean residual life (MRL) of the distribution is given by On the other hand, we analogously discuss the reversed residual life and some of its properties. The reversed residual life can be defined as the conditional random variable − | ≤ which denotes the time elapsed from the failure of a component given that its life is less than or equal to . This random variable may also be called the inactivity time (or time since failure); for more details you may see (Kundu and Nanda, 2010).
Also, in reliability, the mean reversed residual life and ratio of two consecutive moments of reversed residual life characterize the distribution uniquely. The reversed failure (or reversed hazard) rate function is given by Equation (7). The ℎ order moment of the reversed residual life can be obtained by the well known formula Applying the binomial expansion of ( − ) into the above formula gives ]. (21) Using ( ) and 2 ( ) one can obtain the variance and the coefficient of variation of the reversed residual life of the distribution.

Bonferroni and Lorenz Curves
In this subsection we proposed the Bonferroni and Lorenz Curves. The Bonferroni and Lorenz curves (Bonferroni, 1930)  ].

Mean deviation
In statistics, mean deviation about the mean and mean deviation about the median measure the amount of scatter in a population. For random variable with pdf ( ), ].

Maximum Likelihood Estimation
Statistical inference can be carried out in three different ways: point estimation, interval estimation and hypothesis testing. Several approaches for parameter point estimation were proposed in the literature but the maximum likelihood method is the most commonly employed. The enjoy desirable properties and can be used when constructing confidence intervals and regions and also in test statistics. Here, we determine the maximum likelihood estimates ( ) of the parameters of the distribution from complete samples only. Let 1 , . . . , be a random sample of size from the distribution given by (4). Let = ( , , , , ) be 5 × 1 vector of parameters. The total log-likelihood function for is given by The log-likelihood can be maximized either directly by using the SAS program or Rlanguage The maximum likelihood estimation ( ) of , say ̂, is obtained by solving the nonlinear system ( ) = 0. These equations cannot be solved analytically, and statistical software can be used to solve them numerically via iterative methods. We can use iterative techniques such as a Newton-Raphson type algorithm to obtain the estimate ̂. For interval estimation and hypothesis tests on the model parameters, we require the information matrix. The 5 × 5 observed information matrix is given by where is the upper 100 ℎ percentile of the standard normal distribution.

Simulation Study
To test the validating of the theoretical results in Section 5, we produced simulations by These results are reported in Table 2. We can see that the values of | (̂)| and (̂) decrease as sample size increases.

Application
In this section we illustrate the superiority of the new distribution as compared to some of its submodels and also to the alternative distributions. For the data set, the estimates of the parameters of the distributions and information criterion statistics are calculated. Here, we provide application to real data in order to illustrate the potentiality of the model. The measures of goodness-of-fit including the log-likelihood function evaluated at the MLEs, Cramer-von Mises ( * ), Anderson-Darling ( * ) and Kolmogorov-Smirnov ( − ) statistic with its -value are calculated to compare the fitted models. In general, the smallest the values of * , * and − statistics, and the largest − -value, is considered the best fit to the data. The required computations are carried out using the software.