The Extended Burr XII Distribution with Variable Shapes for the Hazard Rate

We define and study a new continuous distribution called the exponentiated Weibull Burr XII. Its density function can be expressed as a linear mixture of Burr XII. Its hazard rate is very flexibile in accomodating various shapes including constant, decreasing, increasing, J-shape, unimodal or bathtub shapes. Various of its structural properties are investigated including explicit expressions for the ordinary and incomplete moments, generating function, mean residual life, mean inactivity time and order statistics. We adopted the maximum likelihood method for estimating the model parameters. The flexibility of the new family is illustrated by means of a real data application.


Introduction
The statistical literature contains many generalized distributions which extensively used in modeling data in various applied areas such as reliability, insurance and life testing.The Burr-XII (BXII) distribution, proposed by Burr (1942), has many generalized forms in the literature such as the beta BXII distribution studied by Paranaíba et  (2017), the odd Lindley BXII distribution due to Abouelmagd et al. (2017) and the Kumaraswamy exponentiated BXII distribution defined by Mead and Afify (2017).
In this paper, we define and study a new five-parameter model named the exponentiated Weibull Burr XII (EWBXII) distribution which extends the WBXII distribution (Afify et al., 2017) and provide greater flexibility in modelling data.Based on the exponentiated Weibull-H (EW-H) family proposed by Cordeiro et al. (2017), we construct our newly model.
Consider the bsaeline cumulative distribution function (CDF) (; ) with parameter vector .Then, the CDF of the EW-H family is given by ( The probability density function (PDF) of the EW-H family is (; , , , ) = ℎ(; ) where ,  and  are three additional positive shape parameters which provide greater flexibility in accommodating all forms of the hazard rate function (HRF).
The EWBXII distribution is a flexible model which exhibit all forms of the hazard rate function (HRF) as shown in Figure 2. Further, the EWBXII distribution contains seventeen sub-models as special cases.
The rest of the paper is outlined as follows.In Section 2, we define the EWBXII distribution, present its special models and provide some plots of its PDF and HRF.In Section 3, we derive a linear representation for the EWBXII PDF and obtain some of its mathematical properties including ordinary and incomplete moments, mean residual life, mean inactivity time, moment generating function (mgf) and order statistics.In Section 4, we use maximum likelihood to estimate the model parameters.In Section 5, we use a real data set to prove empirically the flexibility of the new model.Finally, some concluding remarks are given in Section 6.

The EWBXII distribution
In this section, we define the EWBXII distribution, provide its special cases and give some plots for its PDF and HRF.
By inserting the CDF of the BXII in equation ( 1), we obtain the CDF of the EWBXII distribution and the corresponding PDF of (3) is given by where , , ,  and  are positive shape parameters which can provide more flexibility to model various real world data.
Plots of the PDF and HRF of the EWBXII distribution are displayed in Figures 1 and 2, respectively.The plots in Figure 1 show that the PDF of the EWBXII distribution can be reversed J-shape, right-skewed, left-skewed, symmetric or concave down.Figure 2 shows that the HRF of the proposed model can be constant, decreasing, increasing, J-shape, unimodal, bathtub shapes.

Some EWBXII properties
Some properties of the EWBXII distribution including ordinary and incomplete moments, generating function, mean residual life (MRL), mean inactivity time (MIT) and order statistics are derived in this section.

Linear representation
According to equation (4.5) in Cordeiro et al. (2017), the PDF of the EWBXII distribution can be rewritten as Applying the generalized binomial expansion to the last term, we have where the constant term is given by and  (+1) () refer to the BXII PDF with parameters  and ( + 1).Equation ( 5) is important in deriving some properties of the EWBXII distributions from those of the BXII distribution.

Moments
The th ordinary moment of , follows from (5) (for ( + 1) > ) as The mean of  follows by setting  = 1 in the above equation.
The th incomplete moment of the EWBXII distribution follows from (5) as which is important to calculate the Bonferroni and Lorenz curves and the MRL and MIT.
The MRL or life expectancy at age  is defined by Using  1 (), we obtain The MIT is defined (for  > 0) by By inserting  1 () in the above equation, we have the MIT of  as ( + 1)  (  ; ( + 1) − 1  , 1  + 1).
The mgf of  follows from (5) as where  (+1) () is the mgf of the BXII distribution with two parameters  and ( + 1).

Order statistics
Let  1 , … ,   be a random sample of size  from the EWBXII distribution and let  (1) , … ,  () be the corresponding order statistics.Then, the pdf of the th order statistic,  : , is given by Based on equation (6.
Using the generalized binomial expansion, the last equation reduces to Substituting the last equation in equation ( 8), the pdf of  : comes out as where and  (+1) () as before is the BXII PDF with parameters  and ( + 1).

Estimation
In this section, the unknown parameters of the EWBXII model are estimated via the maximum likelihood from complete samples.Consider a random sample of size , and Respectively, where   =    s  −−1 log  .
The estimate of the unknown parameters can be obtained by setting the score vector to zero, ( ̂) = 0. We can get the MLEs  ̂,  ̂,  ̂,  ̂ and ̂ by solving the above system of equations simultaneously using numerically method with iterative techniques such as the Newton-Raphson algorithm.

Data analysis
In this section, we illustrate the flexibility and importance of the EWBXII distribution using a real data set.The data refer to nicotine measurements, made from several brands of cigarettes in 1998, collected by the Federal Trade.This data set contains  = 346 observations.We shall compare the fit of the EWBXII distribution with some other competitive distributions, namely: the WBXII and beta BXII ( ,  > 0,  > 0,  > 0,  > 0 and δ > 0. The Akaike information criterion (), consistent Akaike information criterion (), Bayesian information criterion (), Hannan-Quinn information criterion () and −ℓ ̂, where ℓ ̂ is the maximized log-likelihood, are used for comparing the fitted models.
Tables 2 and 4 list the values of −ℓ ̂, , ,  and  whereas the MLEs and their corresponding standard errors (in parentheses) of the model parameters are given in Tables 3 and 5.

Conclusions
We study a new five-parameter model called the exponentiated Weibull Burr XII (EWBXII) distribution which generalizes the two-parameter Burr XII distribution and ….We provide some mathematical properties of the new family including explicit expansions for the ordinary and incomplete moments, quantile and generating functions and entropies.The maximum likelihood estimation of the model parameters is investigated.By means of a real data application, we show that the EWBXII distribution can provide better fit than some other well-known models.

Figure 4 :
Figure 4: Fitted PDF of the EWBXII distribution and other fitted PDFs

Table 1
provides all sub-models of the EWBXII distribution.

Table 3 : MLEs and their SEs (in parentheses) InTable 2 ,
we compare the fits of the EWBXII distribution with the BBXII, WBXII, BW, AW, TWL, KMOF and BXII distributions.One can see, from Table1, that the EWBXII distribution has the lowest values for goodness-of-fit statistics among all fitted models.So it could be chosen as the best model for the nicotine data.The histogram and the estimated densities for nicotine data are displayed in Figure3.These plots reveal that the EWBXII distribution is the best model to fit this data set.The fitted PDF, CDF, survival function (SF) and Q-Q plots of the EWBXII distribution are shown in Figure4.