The Flexible Weibull Extension-Burr XII Distribution: Model, Properties and Applications

This paper is devoted to study a new fourparameter additive model. The newly suggested model is referred to as the flexible Weibull extension-Burr XII distribution. It is derived by considering a serial system with one component following a flexible Weibull extension distribution and another following a Burr XII distribution. The usefulness of the model stems from the flexibility of its failure rate which accommodates bathtub and modified bathtub among other risk patterns. These two patterns have been widely accepted in several fields, especially reliability and engineering fields. In addition, the importance of the new distribution is that it includes new submodels which are not known in the literature. Some statistical properties of the proposed distribution such as quantile function, the mode, the rth moment, the moment generating function and the order statistics are discussed. Moreover, the method of maximum likelihood is used to estimate the parameters of the model. Also, to evaluate the performance of the estimators, a simulation study is carried out. Finally, the performance of the proposed distribution is compared through a real data set to some well-known distributions including the new modified Weibull, the additive Burr and the additive Weibull distributions. It is shown that the proposed model provides the best fit for the used real data set.


The Model 2.1. Definition
The failure rate function of a random variable X following the flexible Weibull extension-Burr XII (denoted by

Interpretation of the Failure Rate Curves
The shape of a failure rate function can be characterized by studying the limiting behavior of the failure rate function and its derivative. From Equation (1), differentiating the ℎ( ; Θ) with respect to x gives:  ℎ( ; Θ) = ∞. In this case, ℎ ( ; α , β ) is modified bathtub whereas ℎ ( ; , ) is unimodal.

Sub-models
Three new distributions obtained from the FWBXII distribution as follows

Some Statistical Properties
In this Section, some of the statistical properties including, quantile function, the mode, the median, r th moments, the moment generating function (mgf) and the order statistics are presented as follows.

Mode and Quantile
In this sub-section, the mode(s) and the quantile function of the FWBXII distribution are presented. The mode of a distribution is the value of x corresponding to the maximum value of the probability density function. For a unimodal distribution, x has only one value representing one mode, while for a bimodal distribution x has two values representing two modes. The mode(s) of the FWBXII distribution can be obtained by differentiating the pdf in Equation (2) with respect to x and equating it to zero.
It is difficult to get an explicit solution of Equation (7), therefore, it can be solved numerically. Let X be a random variable with distribution function F as defined in Equation (3), and let J ∈ (0, 1). Then, the quantile L of the FWBXII distribution is obtained as a numerical solution of the following nonlinear Equation with respect to L : It is difficult to get an explicit solution of Equation (8), hence, it can be solved numerically. Special quantiles can be obtained using Equation (8). For example, the median which is the 0.5 quantile can be obtained by setting q = 0.5 in Equation (8). Some values of median and mode(s) for various values of the parameters (α, β, c, k) are calculated in Table 1. From Table 1, we note two cases of modes; unimodal and bimodal.

The Moments
Moments of a distribution can be used to study many important characteristics. So, in this subsection we will present the moments of the FWBXII distribution. The r th moment of the FWBXII random variable X is given by under the condition 0 < UX[ Y < k. Proof: Using integration by parts Using series expansion of , *− c d +, we get Using series expansion of of (W)( ) and (W)( c d ) , we get .
Let e = f Hence Thus; Some of the most important features and characteristics of a distribution can be studied through moments (e.g., tendency, dispersion, skewness and kurtosis). The variance (σ 2 ), coefficient of variation (CV), coefficient of skewness (CS) and coefficient of kurtosis (CK) are given by g = S T − S , The r th moment of the FWBXII can be alternatively obtained using numerical integration. The following table displays the first four moments and the corresponding g , CV, CS and CK values for various choices of the parameters (α, β, c, k). It is observed from this table that by choosing different values of the parameters the distribution can cover both cases of positive and negative skewness.

The Moment Generating Function
If X has FWBXII distribution (Θ) distribution, its moment generating function can be written in the form under the condition 0 < UX[ Y < k. Proof: Using the expression of w Substituting from (9) into (11); (10) is obtained.

The Order Statistics
Order statistics play an important role in probability and statistics. In this subsection, we present the distribution of the i th order statistic from the FWBXII distribution. Let f ( ) ≤ f ( ) ≤ ⋯ ≤ f (z) denote the order statistics obtained from a random sample f1, f2, · · ·, f{ which is taken from the FWBXII distribution with f (x; Θ) and F(x; Θ) given by Equation (2) and Equation (3) Since 0 < 2( )<1 for x>0, by using the binomial expansion theorem for [1 − 2( ; Θ)] z W we obtain Substituting Substituting the pdf and cdf given by (2) and (3) in (14), the pdf u (|) ( ) of the FWBXII distribution is given by

Maximum Likelihood Estimation
In this section, we discuss the estimation of the 4-dimensional parameter vector } = ( , , , ) of the FWBXII distribution by using the method of maximum likelihood. Let f , f , … , f z be a random sample of complete data from the FWBXII distribution defined by (2) and suppose that we are interested in estimating the unknown parameters. The Likelihood function of this sample for Θ = ( , , , ) takes the form Substituting ( W ; Θ) defined by (2) into (16), the above can be written as The corresponding log-likelihood function in this case is then given by By differentiating the log-likelihood function with respect to the parameters (α, β, c, k) and setting the result to zero, we obtain the following system of nonlinear Equations.  (1). It is clear that there is no explicit solution for the above system of nonlinear Equations. Therefore we can use a numerical method such as the Newton-Raphson method to solve these nonlinear Equations and obtain the maximum likelihood estimates of the four parameters, which will be referred to as ", … , ̂ and ‡ . This technique required obtaining the second partial derivatives of the log-likelihood function with respect to the parameters. These derivatives are given in the Appendix.

Data Analysis
In this section, we analyze a real data set to illustrate the applicability and flexibility of the FWBXII distribution for data modeling compared with many known distributions such as additive Weibull (AddW) presented (by Xie and Lai, respectively. We have fitted all selected distributions using the method of maximum likelihood for each data set. In order to compare the aforementioned distributions with the proposed distribution, we applied formal goodness-of-fit tests to verify which distribution fits better to the real data set. Here, we calculated the Kolmogorov Smirnov (K-S) distance test statistic and its corresponding p-value, the log-likelihood (ℒ), Akaike information criterion (AIC), Akaike information criterion with correction (AICc) and Bayesian information criterion (BIC) values, where š›h = 2oe − 2ln (•), š›h = š›h + 2 oe(oe + 1) { − oe − 1 , `›h = oe ln({) − 2 ln(•), • = •M} ‡ N is the value of the likelihood function evaluated at the parameter estimates, n is the number of observations, and m is the number of estimated parameters. In general, the distribution which has the smallest values of these statistics is the better fit for the data.

Data Set
The data corresponds to the time in months to first failure of any kind of small electric cars used for internal transportation and delivery in a large manufacturing facility that were taken from (Nelson, (1982)). The MLEs of the parameters (standard error in parenthesis), log-likelihood, the K-S test statistic with its corresponding p-value, AIC, AICc and BIC are presented in Table 4. The results of this Table indicate that the FWBXII distribution has the lowest K-S value and the highest p-value, which means that the new distribution fits the data better than the other distributions. In addition, our new model has the smallest values of the (AIC, AICc and BIC) statistics. Thus, we can say that the FWBXII is the best among the other fitted models used here. More information is provided by a visual comparison of the histogram of the data with the fitted density functions when the distribution is assumed to be AddW, AddBXII, NMW, EFW, FW and BXII in Figure 3(a). In addition, the comparison of the empirical curve of the reliability function by using the Kaplan-Meier method and the fitted reliability functions is displayed in Figure 3(b). From Figure 3, we can conclude that the FWBXII distribution provides a very good fit for these data compared to all other distributions considered here.

Sub-Models of FWBXII
For the sake of simplicity, we will compare the new three distributions, FWCE, FWCR and FWLogL, with the FWBXII distribution to see if any of these three distributions can perform as well as the FWBXII distribution, since it is better to reduce the number of parameters of any distribution. The MLEs of the parameters (standard error in parenthesis), log-likelihood, the K-S test statistic with its corresponding p-value, AIC, AICc and BIC of FWBXII, FWCE, FWCR and FWLogL are presented in Tables 5. It is clear that the FWBXII, FWCE and FWCR distributions fit the data well. All of them have very small K-S values and high p-value. The FWBXII distribution has the larger log-likelihood value. However, the FWCE distribution has the smaller values for AIC, AICc and BIC. Also, Figure 4 show that the FWCE distribution is nearly as good as the FWBXII distribution. In addition, the likelihood ratio test (LRT) is used to test the reduced model B ' : = 1 against the original model B ž : ≠ 1. The LRT statistic is 0.220 (p-value=0.6390), with one degrees of freedom. So there is no significant evidence to reject B ' . Therefore, we can conclude that reducing the number of parameters to three by fixing one of them still provides a better fit for the used data set.

Conclusion
In this paper, we propose a new distribution called FWBXII distribution. Its definition, the behavior of its failure rate function and some of its statistical properties including the mode, quantile function, moments, moment generating function and order statistics are studied. We use the maximum likelihood method for estimating parameters. Furthermore, the importance of the new distribution is that it includes FWCE, FWCR and FWLogL, which are not known in the literature. In addition, in order to illustrate the usefulness, flexibility and applicability of the distribution, one real data set is analyzed using the new distribution and it is compared with AddW, AddBXII, NMW, EFW, FW and BXII distributions. It is evident from the comparisons that the new distribution is the best distribution for fitting this data set compared to other distributions considered here.