Extended Weibull Burr XII Distribution: Properties and Applications

A new distribution called the Weibull Generalized Burr XII distribution is introduced along with simple physical motivation. The new distribution includes fourteen sub-models, seven of them are new. The new model can be used in modeling bimodal data sets. Set of its properties are derived in details. Two applications are provided to illustrate the importance of the new model. The new model is better that other competitive models via two applications. The method of maximum likelihood is used to estimate the unknown parameters.

where ( ) = (WGBXII) ( ) ( ), ( ) = (WGBXII) ( ) ( ), = ( , , , ), > 0 and > 0 are two additional shape parameters. From Figure 1(a) we conclude that the the PDF of the WGBXII model can be bimodal and left skewed, from Figure 1 The additional parameters and are sought as a manner to furnish a more flexible BXII distribution (see Figure 1). In this work, we study the WGBXII model and give a sufficient description of its mathematical properties. The new model is motivated by its important flexibility in applications (see section. 4), by means of two applications, it is noted that the WGBXII model provides better fits than nine BXII models. The PDF of the WGBXII model can be expressed as Similarly, the CDF (2) of can be expressed in the mixture form where ( ,(1+ ) ) ( ) is the BXII CDF with parameters and (1 + ) . A physical interpretation of the WGBXII distribution can be shown as follows: suppose that we have a lifetime r.v., , having BXII distribution. The generalized ratio that an individual (or component) following the lifetime will die (fail) at time is (1 + ) − 1. Consider that the variability of this ratio of death is represented by the r.v. and assume that it follows the Weibull model with shape . We can write Pr( ≤ ) = Pr( ≤ (1 + ) − 1) = ( ), which is given by (1).

Mathematical properties
The n ℎ ordinary moment of is given by Then, we obtain is the beta function of the second type. By setting = 1 in (5), we get the mean of . The last integration is computed numerically for the new distributions (see Table 1). The skewness and kurtosis measures can be calculated from the ordinary moments using wellknown relationships. The mean, variance, skewness and kurtosis of the WGBXII distribution are computed numerically for = 0.5,1,2,4 and some selected values of , and using the R software. The skewness of the WGLx distribution can range in the interval (−2,86), whereas the kurtosis of the WGBXII distribution varies in the interval (−194,5087.7) also the mean of decreases as increases (see Table 1). The moment generating function (MGF) ( ) = ( ) of can be derived from (3) as The n ℎ incomplete moment ( ( )) of can be expressed from (3) as is the incomplete beta function of the second type, we = 1 we have 1 incomplete moment, the main applications of the 1 incomplete moment refer to the mean deviations and the Bonferroni and Lorenz curves which are very useful in economics, reliability, demography, insurance and medicine.

Probability weighted moments
The (s,r) ℎ PWM of following the WGBXII model, say , , is formally defined by

Order statistics
be a random sample (RS) from the WGBXII distribution and let 1: , … , : be the corresponding order statistics. The PDF of i ℎ order statistic, say : , can be written as where (⋅,⋅) is the beta function. Inserting (1) and (2) in equation (6)

Parameter estimation
Consider the estimation of the unknown parameters ( , , , ) of the WGBXII model from the complete data sets by the maximum likelihood (ML) method. Suppose that 1 , ⋯ , be a RS from the WGBXII model with parameter vector =( , , , ) ⊺ . Then the loglikelihood function (ℓ ( )) for is given by ℓ ( ) = log + log + log + log + ( − 1) ∑ =1 log(1 + ) The above ℓ ( ) in (7) can be maximized numerically via SAS (PROC NLMIXED) or R (optim) or Ox program (via sub-routine MaxBFGS), among others. The components of the score vector, ⊺ are easily to be derived.

Applications
We provide two applications to illustrate the importance, potentiality and flexibility of the WGBXII model. For these data, we compare the WGBXII distribution, with BXII,

5.Conclusions
In this work, we introduced a new distribution called the Weibull Generalized BXII (WGBXII) distribution. We introduced a simple physical motivation for the new model. Set of its properties are derived. Two applications are provided to illustrate the importance of the new model. The new model is better that other nine competitive models via two applications. The method of maximum likelihood is used to estimate the unknown parameters. The new model provide adequate fits as compared to other related models with small values for AIC, BIC, CAIC and HQIC. The new model is much better than the standard Burr XII, beta Burr XII, Kumaraswamy Burr XII, Five parameter Kumaraswamy Burr XII , Topp Leone Burr XII, Marshall-Olkin BXII, beta exponentiated Burr XII, Five parameter beta Burr XII, and Zografos-Balakrishnan Burr XII models in modeling breaking stress and leukaemia data sets.