The Weibull distribution (WD) submitted by Weibull [1] , is an important and popular distribution for modeling lifetime data where the hazard rate function is monotone. Recently, 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 [2] . The Modified Weibull (MW) distribution has been derived by Lai et al. [3] and Sarhan and Zaindin [4] . Moreover the Beta-Weibull (BW) distribution has been studied by Famoye et al. [5] , Beta modified Weibull (BMW) distribution, see Silva et al. [6] and Nadarajah et al. [7] and Kumaraswamy Weibull (KW) distribution, see Cordeiro et al. [8] . Furthermore, many generalizations of the Weibull distribution are investigated like a Generalized modified Weibull (GMW) distribution, Carrasco et al. [9] and Exponentiated modified Weibull extension (EMWE) distribution, Sarhan and Apaloo [10] . Also we can find good review of these models in Pham and Lai [11] and Murthy et al. [12] .

The Flexible Weibull (FWE) distribution (Bebbington et al. [13] ) has many applications in life testing experiments, applied statistics, reliability analysis and clinical studies. For more details on this distribution, see [13] . If is a random variable and it has the Flexible Weibull Extension (FWE) distribution with parameters, then 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

and the hazard rate function is

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, such as the expo-

nential Pareto distribution by AL-Kadim and Boshi [14] , exponential lomax distribution by El-Bassiouny et al. [15] . If is a random variable and is the baseline cumulative distribution function with probability density function and the exponential cumulative distribution function is

By using Equation (5) and replacing the random variable with ratio.

The cdf of exponential generalized distribution is defined by AL-Kadim and Boshi [14] and El-Bassiouny et al. [15]

where is a baseline cdf. Hence the pdf corresponding to Equation (6) is given as

This article is organized as follows. The cumulative function, density function and hazard function of the exponential flexible Weibull extension (EFWE) distribution are defined in Section 2. Some of statistical properties including, quantile function and simulation, the mode, median, the skewness and kurtosis and moments are presented in Sections 3. The moment generating function (mgf) is derived in Sections 4. The order statistics is determined in Section 5. The maximum likelihood estimation of the parameters is obtained in Section 6. Real data sets are analyzed in Section 7. Moreover, we discuss the results and compare it with existing distributions. Finally, we introduce the conclusion of our results.

We define in this section three parameters of the Exponential Flexible Weibull Extension EFWE distribution. Using Equation (2) and Equation (1) in Equation (6) and Equation (7) to obtain 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 Equation (8) is given by

where and are two additional shape parameters.

The survival function, hazard rate function, reversed hazard rate function and cumulative hazard rate function of are given by

respectively, and.

Figures 1-6 display the cdf, pdf, survival function, hazard rate function, reversed hazard rate function and cumulative hazard rate function of the EFWE distribution for some parameter values.

Some statistical properties for the EFWE distribution, such as quantile and simulation

median, the mode, moments, the skewness and kurtosis are given as follows.

The moment generating function (mgf) of the EFWE distribution is given by theorem 2.

Theorem 2. If is a random variable from EFWE distribution, then its moment generating function is

Proof. The moment generating function (mgf) of the random variable with the pdf is

using series expansion of, we obtain

Substituting from Equation (21) into Equation (25) we obtain the moment generating function (mgf) of EFWE distribution in the form

This completes the proof. W

In this section, we derive closed form expressions for the probability density function pdf of the order statistic of the EFWE distribution. Suppose denote the order statistics obtained from a random sample which taken from a continuous population with cumulative distribution function and probability density function, then the pdf of is as follows

where and are the probability density function and cumulative distribution function of EFWE distribution given by Equation (9) and Equation (8) respectively, is the Beta function and, also we define first order statistics, and the last order statistics as, where for all. Using the binomial expansion of

Substituting from Equation (27) into Equation (26), we have

Substituting from Equation (8) and Equation (9) into Equation (28), we obtain

Relation (29) shows that is the weighted average of the Exponential Flexible Weibull Extension distribution with different shape parameters.

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.

In this application, we will analysis a real data set given by Aarset [22] , see Table 2, using the EFWE distribution and compare it with the other fitted distributions like a flexible Weibull extension (FWE) distribution, Weibull (W) distribution, linear failure rate (LFR) distribution, Exponentiated Weibull (EW) distribution, generalized linear failure rate (GLFR) distribution and Exponentiated Flexible Weibull (EFW) distribution by using Kolmogorov Smirnov (K-S) statistic, P-value, Akaike information criterion (AIC), as well as Akaike Information Criterion with correction (AICC), see [20] and also Bayesian information criterion (BIC), see [21] , values.

Table 3 gives the maximum likelihood estimations of parameters for EFWE distribution, the value of K-S Statistics and P-value. As well the values of the log-likelihood functions, AIC, AICC and BIC are in Table 4.

From Table 3 and Table 4 we note that the EFWE distribution with three parameters gives a better fit than the previous models distributions. It has the largest log-likelihood

function, and also P-value, on the other hand the smallest K-S, AIC, AICC and BIC values from among those considered in this article.

Substituting the maximum likelihood estimations of the unknown parameters and into Equation (35), we have the estimation of the variance covariance matrix as follows

The approximate 95% two sided confidence intervals of the unknown parameters and are, and, respectively.

From, Figure 7 and Figure 8 we note that the likelihood function have unique solution.

Figure 9 represents the estimation for the survival function, by using the Kaplan-Meier method and its fitted parametric estimations when the distribution is assumed to be FW, W, LFR, EW, GLFR, EFW and EFWE are computed and plotted in the following shape.

Figure 10 and Figure 11 give the form of the hazard rate and cumulative distribution function for the FW, W, LFR, EW, GLFR, EFW and EFWE which are used to fit the data when the unknown parameters included in each distribution are replaced by their maximum likelihood estimation.

We propose a new distribution, based on the exponential generalized distribution. The general idea is to add parameter to a flexible Weibull extension FWE distribution, this new distribution is called the exponential flexible Weibull extension EFWE. Its definition and some of statistical properties are studied. We use the maximum likelihood method for estimating parameters. Finally, the advantage of the EFWE distribution is concluded by an application using real data. Moreover, it is shown that the exponential flexible Weibull extension EFWE distribution fits better than existing modifications of the Weibull and flexible Weibull distributions.

El-Desouky, B.S., Mustafa, A. and Al-Garash, S. (2017) The Exponential Flexible Weibull Extension Dis- tribution. Open Journal of Modelling and Simulation, 5, 83-97. http://dx.doi.org/10.4236/ojmsi.2017.51007