An Extended Burr XII Distribution : Properties , Inference and Applications

We propose and study a new continuous model named the Marshall-Olkin exponentiated Burr XII (MOEBXII) distribution. It contains several special cases, namely the Marshall-Olkin exponentiated loglogistic, Marshall-Olkin exponentiated Lomax, Marshall-Olkin Burr XII, Marshall-Olkin log-logistic, Marshall-Olkin Lomax distributions, among others, and most importantly includes all four of the most common types of hazard function: monotonically increasing or decreasing, bathtub and arc-shaped hazard functions. Some of its structural properties are obtained such as the ordinary and incomplete moments, quantile and generating functions, order statistics and probability weighted moments. The maximum likelihood and least square methods are used to estimate the model parameters. A simulation study is performed to evaluate the precision of the estimates from both methods. The usefulness of the new model is illustrated by means of two real data sets.


Introduction
The Burr XII (BXII) distribution includes as sub-models the logistic and Weibull distributions and it is very popular for modeling lifetime data and phenomenon with different failure rates.It can fit a wide range of empirical data.Different values of its parameters cover a broad set of skewness and kurtosis.Hence, it is used in various fields such as finance, hydrology and reliability to model a variety of data types.Examples of data modeled by the BXII distribution are household income, crop prices, insurance risk, flood levels and failure data.
Recently, many generalizations of the BXII distribution have been proposed.For example, the beta BXII (Paranaba et al., 2011), Kumaraswamy BXII (Paranaba et al., In this paper, the EBXII distribution is embedded in a larger family by adding a new shape parameter.The model generated by applying the MO transformation to the EBXII distribution is called the Marshall-Olkin exponentiated Burr XII (MOEBXII) distribution.The new distribution has the advantage of being capable of modelling various shapes of aging and failure criteria.Further, the MOEBXII model is shown to provide better fits than at least seven other models each one having the same number of parameters.We hope that the proposed distribution will attract wider applications in reliability, engineering and other areas of research.
The rest of the paper is outlined as follows.In Section 2, we define the MOEBXII distribution, provide plots of the pdf and hrf, give some special models and derive a linear representation for the pdf.In Section 3, we obtain some mathematical properties including quantile function (qf), ordinary, central and incomplete moments, moment generating function (mgf), moments of the residual and reversed residual lifes, order statistics and probability weighted moments (PWMs).In Section 4, we discuss the maximum likelihood and the least square methods to estimate the model parameters.Two simulation studies are performed in Section 5 in order to verify the consistency of the estimates.In Section 6, we prove empirically the potentiality of the new distribution by means of two real data sets.Finally, some concluding remarks are offered in Section 7.

The MOEBXII Distribution
In this section, we define the MOEBXII distribution and provide its eleven sub-models.By inserting (1) in equation (3), the cdf of the MOEBXII distribution (for  > 0), say () = (; , , , ), is given by where , ,  and  are positive shape parameters.
The hrf and cumulative hazard rate function (chrf) of  are given by ℎ(; , , , ) = (; , , ) respectively, where For 0 <  < 1, we provide a simple motivation for the MOEBXII distribution in the medical context as follows.Consider a random sample  1 , … ,   , where the   's are independent and identically distributed having the EBXII density (2).Let  have a geometric random variable with probability mass function (pmf) ( = ) =  (1 − ) −1 (for  ≥ 1).Then, the density of the random variable  = min( 1 , … ,   ) is that of (7).For  > 1, we can use  −1 instead of  in the pmf of .This setup is usually common in oncology, where  represents the amount of cells with metastasis potential and   denotes the time for the th cell to metastasize.So,  represents the recurrence time of the cancer.
The MOEBXII distribution is a very flexible model that approaches to different distributions.Its eleven sub-models are listed in Table 1. Figure 1 displays some plots of the MOEBXII density for selected values of , ,  and .These plots illustrate the versatility and modality of the new distribution.The plots in Figure 2 reveal that the hrf of  can have bathtub, unimodal, increasing and decreasing shapes.Abbreviations: Lo=Lomax and LL=Log-Logistic.

Linear Representation
An expansion for equation ( 7) can be derived using the power series where || < 1,  > 0 and Γ() = ∫ ∞ 0  −1 e − d is the gamma function.Based on the linear representation of Cordeiro et al. (2014) for the MO-G family and using (8) in (7) when  ∈ (0,1), we can write Using the generalized binomial expansion, we obtain Then, the pdf of  can be expressed as a linear combination of BXII densities where  +1 () is the pdf of the BXII model with shape parameters  and ( + 1) and (for  ≥ 0) Otherwise, if  > 1, we can rewrite (7), using (8) and following the same developments of Cordeiro et al. (2014), as Applying the generalized binomial expansion gives Then, the pdf of  when  > 0 can be expressed as a linear combination of BXII densities given by where Equations ( 9) and (10) have the same linear representation except for the coefficients.They are the main results of this section.So, some structural properties of the MOEBXII model can be determined from these equations and those properties of the BXII distribution.From now on, we work only with equation (9).

Mathematical Properties
In this section, we obtain some mathematical properties of the MOEBXII distribution including qf and random number generation, ordinary, central and incomplete moments, mgf and moments of the residual and reversed residual lifes.
Let  be a random variable having the two-parameter BXII density given by where  and  are positive shape parameters.−1)  are the beta and incomplete beta functions, respectively.

Moments
The th ordinary moment of  is given by For  <  , we obtain The mean of  follows by setting  = 1 in (12) as ).
The th central moment (  ) of  is given by The skewness and kurtosis measures can be calculated from the central moments using well-known relationships.
The th incomplete moment of  is given by   () = ∫  0   ().It follows from equation ( 9) and then using the incomplete beta function, we obtain (for  < ) The first incomplete moment of  follows from the last equation by setting  = 1.It can be applied to obtain mean deviations, Bonferroni and Lorenz curves, mean residual and waiting times and totality of deviations from the mean and median.
The effects of the parameter  on the mean, variance, skewness and kurtosis for given values of ,  and  are displayed in Figures 3 and 4, respectively.

Quantile and Generating Functions
The qf of  is obtained by inverting (6) as Setting  = 0.5 in (13) gives the median of .Simulating the MOEBXII random variable is straightforward.If  is a uniform variate on the unit interval (0,1), then the random variable  = () follows (7).
First, we provide the mgf of the two-parameter BXII distribution as defined by Afify et al. (2016).We can write the mgf of  with pdf (11), say   (), (for  < 0) as We require the Meijer G-function defined by where  = √−1 is the complex unit and  denotes an integration path (Gradshteyn and Ryzhik, 2000, Section 9.3).The Meijer G-function contains as particular cases many integrals with elementary and special functions (Prudnikov et al., 1986).Further, we assume that  = /, where  and  are positive integers.This condition is not restrictive since every positive real number can be approximated by a rational number. .For  < 0, we can write Finally, the mgf of  can be reduced to , −( + 1) − 1).
The mean residual life (MRL) function of  follows by setting  = 1 in the last equation.It represents the expected additional life length for a unit which is alive at age .The MRL is also known as the life expectancy at age .
For  = 1,2, … and  > 0, the th moment of the reversed residual life of  is given by Then, we can write The mean inactivity time (MIT) of  follows by setting  = 1 in the above equation.It represents the waiting time elapsed since the failure of an item on condition that this failure had occurred in (0, ).The MIT is also called the mean reversed residual life function.

Order Statistics
Let  1 , … ,   be a random sample of size  from the MOEBXII distribution and  (1) , … ,  () be the corresponding order statistics.Then, the pdf of the th order statistic  : , say  : (), is given by Using the generalized binomial expansion and after some algebra, we can write Using this expansion again, the last equation reduces to Then, we can write where and  +1 () denotes the BXII density function with parameters  and ( + 1).
By inserting (15) in equation ( 14), we obtain Equation ( 16) reveals that the density function of the MOEBXII order statistics is a linear combination of BXII densities.Based on this equation, we can obtain some structural properties of  : from those BXII properties.For example, the th moment of  : is given by Using the moments in the last equation, we can derive explicit expressions for the Lmoments of  as infinite weighted linear combinations of the means of suitable MOEBXII distributions.They are given by ) ( −: ),  ≥ 1.

Probability Weighted Moments
The PWMs can be used to derive estimators of the parameters and quantiles of generalized distributions.These moments have low variances and no severe biases, and they compare favorably with estimators obtained by maximum likelihood.The (, )th PWM of  (for  ≥ 1,  ≥ 0) is formally defined by We can write from ( 9) We can rewrite the last equation as where Then,  , can be expressed as

Estimation
In this section, we consider the estimation of the unknown parameters of the MOEBXII model from complete samples by maximum likelihood and least square methods.The MLEs enjoy desirable properties for constructing confidence intervals.
Let  1 , … ,   be a random sample of this distribution with parameter vector  = (, , , ) T .The log-likelihood function for , say ℓ = ℓ(), is given by where Equation ( 18) can be maximized either directly by using the R (optim function), SAS (PROC NLMIXED), Ox program (sub-routine MaxBFGS) or by solving the nonlinear likelihood equations obtained by differentiating this equation.We can obtain the estimates of the unknown parameters by setting the score vector to zero, ( ̂) = .By solving these equations simultaneously gives the MLEs ,  ̂,  ̂ and  ̂.
Iterative techniques such as Newton-Raphson type algorithms can be adopted to obtain the estimates.All the second-order derivatives exist for the proposed distribution.
An alternative estimation to maximum likelihood is the least square estimation.The least square estimates (LSEs), ,  ̃,  ̃ and  ̃ of , ,  and , are defined as those arguments that minimize the objective function: where  () is a possible outcome of the th order statistic based on a -points random sample obtained from .For more complex distributions, the properties of the LSEs are not easy to obtain as those of the MLEs.
The minimum point ,  ̃,  ̃ and  ̃ can also be given as a solution in the following system of non-linear equations:

Simulation Study
In this section, we provide simulation results to assess the performance of the MLEs and LSEs of.We simulate the MOEBXII model under two setups for the model parameters:  = 1.5,  = 2.0,  = 1.5 and  = 0.5; and  = 3.0,  = 0.5,  = 1.5 and  = 5.0.
The results are obtained from 3,000 Monte Carlo simulations.For each replication, a random sample of size  is drawn from the MOEBXII model and the parameters are estimated by MLEs.The sample sizes are  = 50, 100, 200, 300 and 500.All simulations were performed using the R software with the Optim function.
Table 2 lists the averages of the MLEs (Mean), the biases (Bias) and the mean square errors (MSEs) and 95% coverage probabilities (CP).We conclude from the figures in Table 2 that the MSEs of the MLEs of , ,  and  decay toward zero when the sample size increases, as expected under standard asymptotic theory.In fact, the biases of the estimates tend to be closer to the true parameter values if  increases.
Table 3 presents the averages of the LSEs (Mean), the biases (Bias) and the mean square errors (MSEs).We conclude from the figures in Table 3 that the biases and MSEs of the LSEs of , ,  and  are, in general, lower than those values for the MLEs.The parameters of the above densities are all positive real numbers.
In order to compare these distributions, we consider some goodness-of-fit measures including −2ℓ ̂ (maximized log-likelihood),  (Akaike information criterion),  (Bayesian information criterion),  (consistent Akaike information criterion), Kolmogorov-Smirnov (K-S) statistic and -value.4 and 6 list the values of −2ℓ ̂, , , , K-S and -value, whereas the MLEs and their corresponding standard errors (in parentheses) of the model parameters are given in Tables 5 and 7, for the two data sets, respectively.
The fitted pdfs of the MOEBXII, KwL, WL and BW models and QQ-plots for cancer data are displayed in Figure 5, whereas Figure 6 displays the estimated cdfs and survival functions of these fitted models.Overall, the values of the goodness-of-fit statistics in Tables 4 and 6 indicate that the MOEBXII model has the lowest values of these statistics among all fitted competitive models.Moreover, the plots in Figures 5, 6, 7 and 8 reveal that this model gives a close fit to both data sets.So, the MOEBXII distribution could be chosen as the best model to fit both data sets.

Conclusions
In this paper, we propose a new four-parameter model called the Marshall-Olkin exponentiated Burr XII (MOEBXII) distribution, which extends the exponentiated Burr XII (EBXII) distribution.The MOEBXII density function can be expressed as a linear combination of BXII densities.We derive explicit expressions for the ordinary and incomplete moments, quantile and generating functions, residual and reversed residual life functions.We also obtain the density function of the order statistics and their moments.We discuss maximum likelihood and least squares estimation.The proposed distribution provides better fits than some other nested and non-nested models using two real data sets.We hope that the proposed model will attract wider applications in areas such as survival and lifetime data, meteorology, hydrology, reliability, engineering and others

Figure 1 :
Figure 1: Plots of the MOEBXII density for selected parameter values.

Figure 2 :
Figure 2: Plots of the MOEBXII hrf for selected parameter values.

Figure 3 :
Figure 3: Plots of mean and variance of the MOEBXII distribution.

Figure 4 :
Figure 4: Plots of skewness and kurtosis of the MOEBXII distribution.

Figure 5 :
Figure 5: The estimated MOEBXII pdf and other estimated pdfs (left panel) and the QQplots of the MOEBXII, KwL, WL and BW models (right panel) for cancer data.

Figure 7
Figure7displays the fitted pdfs of the MOEBXII, KwLL, BBXII and BW distributions and QQ-plots for relief times data.Figure8gives the plots of the estimated cdfs and survival functions for the fitted models.

Figure 6 :
Figure 6: The estimated cdf of the MOEBXII model and other estimated cdfs (left panel) and estimated survival function of the MOEBXII model and other estimated survival functions (right panel) for cancer data.

Figure 7 :
Figure 7: The estimated MOEBXII pdf and other estimated pdfs (left panel) and the QQplots of the MOEBXII, KwLL, BBXII and BW models (right panel) for relief times data.

Figure 8 :
Figure 8: The estimated cdf of the MOEBXII model and other estimated cdfs (left panel) and estimated survival function of the MOEBXII model and other estimated survival functions (right panel) for relief times data.

Table 3 : Summaries of the LSEs for the MOEBXII model
In this section, we provide two applications of the new distribution to prove empirically its potentiality.We shall compare the fits of the MOEBXII model and the following competing non-nested distributions, namely: the Weibull Lomax (WL) (Tahir et al., 2015), beta generalized exponential (BGE)(Cordeiro et al., 2010), beta Burr XII (BBXII)