The Odd Exponentiated Half-Logistic Burr XII Distribution

A new lifetime model called the odd exponentiated half-logistic Burr XII distribution is defined and studied. Its density function can be expressed as a linear mixture of Burr XII densities. The proposed model is capable of modeling various shapes of hazard rate including decreasing, increasing, decreasingincreasing-constant, reversed J-shape, J-shape, unimodal or bathtub shapes. Various of its structural properties are investigated. The maximum likelihood method is adopted to estimate the model parameters. The flexibility of the new model is proved empirically using two real data sets. It can serve as an alternative model to other lifetime distributions in the existing literature for modeling positive real data in many areas.


Introduction
The statistical literature contains hundreds of distributions which have several applications in various applied areas such as reliability, engineering, economics, insurance, life testing and biomedical sciences, among other.These applications have indicated that there are many data sets following the classical models are more often the exception rather than the reality.Since, a significant progress has been made towards the generalization of some classical distributions and their successful applications to problems in these areas.
The Burr XII (BXII) distribution (Burr, 1942) with two positive shape parameters,  and , has the cumulative distribution function (CDF) and probability density function (PDF) given (for  > 0) by (; , ) = 1 − (1 +   ) − and (; , ) =  −1 (1 +   ) −−1 (1) The statistical literature contains several generalized forms of the BXII model such as the beta BXII due to Paranaíba et al. (2011), the Kumaraswamy BXII due to Paranaíba et al. (2013), the beta exponentiated BXII due to Mead (2014) (2) The corresponding PDF of ( 2) is given by where (; ) is a bsaeline PDF and  and  are positive shape parameters which provide more flexibility in accommodating all forms of the hazard rate function (HRF) of the generated model.Now, we define the OEHLBXII distribution and provide some plots for its PDF and HRF.The CDF of the OEHLBXII distribution follows, by inserting the CDF (1) in Equation ( 2), as The PDF of the OEHLBXII distribution reduces to (; , , , ) = where , ,  and  are positive shape parameters which can provide more flexibility to model various data in areas such as survival and lifetime data, engineering, income inequality and others.
The OEHLBXII distribution exhibits all important forms of the HRF including J-shape, reversed J-shape, decreasing, increasing, decreasing-increasing-constant, unimodal or bathtub hazard rate shapes.
The PDF and HRF plots of the OEHLBXII distribution are displayed in Figures 1 and 2, respectively.Figure 1 reveals that the PDF of the OEHLBXII distribution can be reversed J-shape, symmetric, concave down right-skewed or left-skewed.The HRF of the OEHLBXII model can be J-shape, reversed J-shape, decreasing, increasing, unimodal or bathtub hazard rate shapes.The rest of the paper is outlined as follows.Section 2, is devoted to derive some mathematical properties of the OEHLBXII distribution.In Section 3, we use maximum likelihood to estimate the model parameters.Two real data sets are analyzed to prove the flexibility of the OEHLBXII model in Section 4. Finally, some concluding remarks are presented in Section 5.

The OEHLBXII properties
Some properties of the OEHLBXII distribution including linear representation, quantile functoin (QF), ordinary and incomplete moments, moment generating function (MGF), mean residual life (MRL), mean inactivity time (MIT) and order statistics are derived in this section.

Linear representation
Using Equation (8) in Afify et al. (2017), the PDF of the OEHLBXII distribution can be expressed as where is the exponentiated BXII density with power parameter ( +  + 1).
Let  be a random variable having the distribution in Equation ( 1).The th ordinary and incomplete moments of are, respectively, given (for  < ) by are, respectively, the beta and the incomplete beta functions of the second type.

Some moments
The th ordinary moment of , follows from (7) (for ( + 1) > ) as The mean of  follows by setting  = 1 in the above equation.
The mean, variance, skewness and kurtosis for different values of , ,  and  are calculated in Table 1.

Order statistics
Let  1 , … ,   be a random sample of size  from the OEHLBXII distribution and let  (1) , … ,  () be the corresponding order statistics.Then, the pdf of the th order statistic, denoted by  : , is given by where  = !/( − 1)! ( − )!.
Using Equation (20) in Afify et al. (2017), one can write After applying the generalized binomial expansion, the last equation can be expressed as By combining the above equation and Equation (10), the PDF of  : reduces to where and  (+1) denotes to the BXII PDF with parameters  and ( + 1).

Estimation
The unknown parameters of the OEHLBXII distribution are estimated using the maximum likelihood from complete samples only.Consider a random sample of size , The estimates of the unknown parameters can be obtained by setting the score vector to zero, ( ̂) = 0. We can get the MLEs  ̂ by solving the above system of equations simultaneously using numerical method with iterative techniques such as the Newton-Raphson algorithm.

Real data applications
In this section, the flexibility and importance of the OEHLBXII distribution are illustrated via two real data sets.The first data set consists of 63 observations of the strengths of 1.5 cm glass fibres, originally obtained by workers at the UK National Physical Laboratory (Smith and Naylor, 1987)

Conclusions
We study a new four-parameter model called the odd exponentiated half-logistic Burr XII (OEHLBXII) distribution which generalizes the two-parameter Burr XII distribution.We provide some mathematical properties of the new model including explicit expansions for the quantile function, ordinary and incomplete moments, mean residual life, mean inactivity time and order statistics.The maximum likelihood estimation of the model parameters is investigated.We prove emprically, via two real data applications, that the OEHLBXII distribution can provide better fits than some other well-known competitive models.
, the Marshall-Olkin extended BXII due to Al-Saiarie et al. (2014), the McDonald BXII due to Gomes et al. (2015), the exponentiated Burr XII Poisson due to da Silva et al. (2015), the Kumaraswamy exponentiated BXII due to Mead and Afify (2017), the Weibull BXII due to Afify et al. (2018) and the odd Lindley BXII due to Abouelmagd et al. (2018).In this paper, we study a new extension of the BXII model called the odd exponentiated half-logistic Burr XII (OEHLBXII) distribution which provides more flexibility in modelling data in several areas.The new model is constructed based on the odd exponentiated half-logistic-G (OEHL-G) family defined by Afify et al. (2017).Let (; ) be a bsaeline CDF with parameter vector .Then, the CDF of the OEHL-G class is defined (for  ∈ ℜ) by (; , , )

Figure 1 :Figure 2 :
Figure 1: Some possible shapes for the PDF of the OEHLBXII distribution is the MGF of the BXII distribution with two parameters  and ( + 1).Paranaíba et al. (2011) provided a simple formula for the MGF of BXII distribution with two-parameter  and  (for  < 0) as () =   (−,

Figure 3 :
Figure 3: Fitted PDF of the OEHLBXII distribution and other fitted PDFs for glass fibres data

Figure 4 :Figure 5 :
Figure 4: Fitted PDF of the OEHLBXII distribution and other fitted PDFs for cancer data

Table 3 : MLEs (their standard errors in parentheses
The histogram and the estimated densities for both data sets are displayed in Figures3 and 4.These plots reveal that the OEHLBXII distribution is the best model to fit both data sets.The fitted PDF, CDF, survival function (SF) and PP plots of the OEHLBXII distribution for both data sets are shown in Figures5 and 6, respectively.
),  * and  * for cancer data