The Exponential Flexible Weibull Extension Distribution

This paper is devoted to study a new three- parameters model called the Exponential Flexible Weibull extension (EFWE) distribution which exhibits bathtub-shaped hazard rate. Some of it's statistical properties are obtained including ordinary and incomplete moments, quantile and generating functions, reliability and order statistics. The method of maximum likelihood is used for estimating the model parameters and the observed Fisher's information matrix is derived. We illustrate the usefulness of the proposed model by applications to real data.


Introduction
The Weibull distribution (WD) introduced by Weibull [23], is a popular distribution for modeling lifetime data where the hazard rate function is monotone. Recently appeared new classes of distributions were based on modifications of the Weibull distribution (WD) to provide a good fit to data set with bathtub hazard failure rate Xie and Lai [21]. Among of these, Modified Weibull (MW) distribution, Lai et al. [9] and Sarhan and Zaindin [17], Beta-Weibull (BW)distribution, Famoye et al. [6], Beta modified Weibull (BMW)distribution, Silva et al. [20] and Nadarajah et al. [14], Kumaraswamy Weibull (KW) distribution, Cordeiro et al. [5], Generalized modified Weibull (GMW) distribution, Carrasco et al. [4] and Exponentiated modified Weibull extension (EMWE) distribution, Sarhan and Apaloo [18], among others. A good review of these models is presented in Pham and Lai [15] and Murthy et al. [13]. The Flexible Weibull (FWE) distribution, Bebbington et al. [3] has a wide range of applications including life testing experiments, reliability analysis, applied statistics and clinical studies. The origin and other aspects of this distribution can be found in [3]. A random variable X is said to have the Flexible Weibull Extension (FWE) distribution with parameters α, β > 0 if it's probability density function (pdf) is given by while the cumulative distribution function (cdf) is given by The survival function is given by the equation x , x > 0, (1.3) and the hazard function is h(x) = (α + β x 2 )e αx− β x . (1.4) In this article, a new generalization of the Flexible Weibull Extension (FWE) distribution called exponential flexible Weibull extension (EFWE) distribution is derived. Using the exponential generator applied to the odds ratio 1 1−G(x) , such as the exponential Pareto distribution by AL-Kadim and Boshi [7], exponential lomax distribution by El-Bassiouny et al. [10]. If G(x) is the baseline cumulative distribution function (cdf) of a random variable, with probability density function (pdf) g(x) and the exponential cumulative distribution function is (1.5) Based on this density, by replacing x with ratio 1 1−G(x) . The cdf of exponential generalized distribution is defined by (see AL-Kadim and Boshi [7] and El-Bassiouny et al. [10]) where G(x) is a baseline cdf. Hence the pdf corresponding to Eq. (1.6) is given by .
This paper is organized as follows, we define the cumulative, density and hazard functions of the exponential flexible Weibull extension (EFWE) distribution in Section 2. In Sections 3 and 4, we introduce the statistical properties including , quantile function skewness and kurtosis, rth moments and moment generating function. The distribution of the order statistics is expressed in Section 5. The maximum likelihood estimation of the parameters is determined in Section 6. Real data sets are analyzed in Section 7 and the results are compared with existing distributions. Finally, we introduce the conclusions in Section 8.

The Exponential Flexible Weibull Extension Distribution
In this section we study the three parameters Exponential Flexible Weibull Extension (EFWE) distribution. Using G(x) Eq. (1.2) and g(x) Eq. (1.1) in Eq. (1.6) and Eq. (1.7) to obtained the cdf and pdf of EFWE distribution. The cumulative distribution function cdf of the Exponential Flexible Weibull Extension distribution (EFWE) is given by The pdf corresponding to Eq. (2.1) is given by where x > 0 and , α, β > 0 are two additional shape parameters. The survival function S(x), hazard rate function h(x), reversed-hazard rate function r(x) and cumulative hazard rate function H(x) of X ∼ EF W E(α, β, λ) are given by respectively, x > 0 and α, β, λ > 0.

Statistical Properties
In this section, we study the statistical properties for the EFWE distribution, specially quantile function and simulation median, skewness, kurtosis and moments.

Quantile and simulation
The quantile x q of the EFWE (α, β, λ) distribution random variable is given by Using the distribution function of EFWE distribution, from (2.1), we have So, the simulation of the EFWE distribution random variable is straightforward. Let U be a uniform variate on the unit interval (0, 1). Thus, by means of the inverse transformation method, we consider the random variable X given by Since the median is 50 % quantile then by setting q = 0.5 in Eq. (3.2), can be obtained the median M of EFWE distribution.

The Mode of EFWE
In this subsection, we will derive the mode of the EFWE(α, β, λ) distribution by deriving its pdf with respect to x and equal it to zero thus the mode of the EFWE(α, β, λ) distribution can be obtained as a nonnegative solution of the following nonlinear equation From Figure ??, the pdf for EFWE distribution has only one peak, It is a unimodal distribution, so the above equation has only one solution. It is not possible to get an explicit solution of (3.5) in the general case. Numerical methods should be used such as bisection or fixed-point method to solve it. Some values of median and mode for various values of parameters α, β and λ calculated in Table 1.

The Skewness and Kurtosis
The analysis of the variability Skewness and Kurtosis on the shape parameters α, β can be investigated based on quantile measures. The short comings of the classical Kurtosis measure are well-known. The Bowely's skewness based on quartiles is given by, Kenney and Keeping [8] where q (.) represents quantile function.

The Moments
In this subsection we discuss the rth moment for EFWE distribution. Moments are important in any statistical analysis, especially in applications. It can be used to study the most important features and characteristics of a distribution (e.g. tendency, dispersion, skewness and kurtosis).
Theorem 3.1. If X has EFWE (α, β, λ) distribution, then the rth moments of random variable X, is given by the following Proof. We start with the well known distribution of the rth moment of the random variable X with probability density function f (x) given by Substituting from Eq. (2.2) into Eq. (3.9) we get then we get using series expansion of e (i+1)e αx− β x , we obtain using series expansion of e −(j+1) β x , by using the definition of gamma function ( Zwillinger [22]), in the form, Finally, we obtain the rth moment of EFWE distribution in the form This completes the proof.

The Moment Generating Function
The moment generating function (mgf) M X (t) of a random variable X provides the basis of an alternative route to analytic results compared with working directly with the pdf and cdf of X.
Theorem 4.1. The moment generating function (mgf) of EFWE distribution is given by (4.1) Proof. The moment generating function of the random variable X with probability density function f (x) is given by using series expansion of e tx , we obtain

Substituting from Eq. (3.8) into Eq. (4.3) we obtain the moment generating function (mgf) of EFWE distribution in the form
This completes the proof.

Order Statistics
In this section, we derive closed form expressions for the probability density function of the rth order statistic of the EFWE distribution. Let X 1:n , X 2:n , · · · , X n:n denote the order statistics obtained from a random sample X 1 , X 2 , · · · , X n which taken from a continuous population with cumulative distribution function cdf F (x; ϕ) and probability density function pdf f (x; ϕ), then the probability density function of X r:n is given by where f (x; ϕ), F (x; ϕ) are the pdf and cdf of EFWE (α, β, λ) distribution given by Eq. (2.2) and Eq. (2.1) respectively, ϕ = (α, β, λ) and B(., .) is the Beta function, also we define first order statistics X 1:n = min(X 1 , X 2 , · · · , X n ), and the last order statistics as X n:n = max(X 1 , X 2 , · · · , X n ). Since 0 < F (x; ϕ) < 1 for x > 0, we can use the binomial expansion of [1 − F (x; ϕ)] n−r given as follows Substituting from Eq. (5.2) into Eq. (5.1), we obtain Substituting from Eq. (2.1) and Eq. (2.2) into Eq. (5.3), we obtain Relation (5.4) shows that f r:n (x; ϕ) is the weighted average of the Exponential Flexible Weibull Extension distribution withe different shape parameters.

Parameters Estimation
In this section, point and interval estimation of the unknown parameters of the EFWE distribution are derived by using the method of maximum likelihood based on a complete sample.

Maximum likelihood estimation
Let x 1 , x 2 , · · · , x n denote a random sample of complete data from the EFWE distribution. The Likelihood function is given as substituting from (2.2) into (6.1), we have The log-likelihood function is The maximum likelihood estimation of the parameters are obtained by differentiating the log-likelihood function L with respect to the parameters α, β and λ and setting the result to zero, we have the following normal equations.

Asymptotic confidence bounds
In this section, we derive the asymptotic confidence intervals when α, β > 0 and λ > 0 as the MLEs of the unknown parameters α, β > 0 and λ > 0 can not be obtained in closed forms, by using variance covariance matrix I −1 see Lawless [11], where I −1 is the inverse of the observed information matrix which defined as follows The second partial derivatives included in I are given as follows.
We can derive the (1 − δ)100% confidence intervals of the parameters α, β and λ, by using variance matrix as in the following formŝ where Z δ 2 is the upper ( δ 2 )-th percentile of the standard normal distribution.

Application
In this section, we present the analysis of a real data set using the EFWE (α, β, λ) model and compare it with the other fitted models like A flexible Weibull extension (FWE) distribution, Weibull distribution (WD) , linear failure rate distribution (LFRD), exponentiated Weibull distribution(EWD), generalized linear failure rate distribution (GLFRD) and exponentiated flexible Weibull distribution (EFWD) using Kolmogorov Smirnov (K-S) statistic, as well as Akaike information criterion (AIC), [2], Akaike Information Citerion with correction (AICC) and Bayesian information criterion (BIC) [19] values. Consider the data have been obtained from Aarset [1], and widely reported in many literatures. It represents the lifetimes of 50 devices, and also, possess a bathtub-shaped failure rate property, Table 2.  Table 3 gives MLEs of parameters of the EFWE distribution and K-S Statistics. The values of the loglikelihood functions, AIC, AICC and BIC are in Table 4.  We find that the EFWE distribution with the three-number of parameters provides a better fit than the previous new modified a flexible Weibull extension distribution(FWE) which was the best in Bebbington et al. [3]. It has the largest likelihood, and the smallest K-S, AIC, AICC and BIC values among those considered in this paper.
Substituting the MLE's of the unknown parameters α, β, λ into (6.6), we get estimation of the variance covariance matrix as the following   The nonparametric estimate of the survival function using the Kaplan-Meier method and its fitted parametric estimations when the distribution is assumed to be F W, W, LF R, EW, GLF R, EF W and EF W E are computed and plotted in Figure 9.

Conclusions
A new distribution, based on exponential generalized distributions, has been proposed and its properties are studied. The idea is to add parameter to a flexible Weibull extension distribution, so that the hazard function is either increasing or more importantly, bathtub shaped. Using Weibull generator component, the distribution has flexibility to model the second peak in a distribution. We have shown that the exponential flexible Weibull extension EFWE distribution fits certain well-known data sets better than existing modifications of the exponential generalized family of probability distribution.