A New Extension of the Burr Type XII Distribution

Email: refaie_top@yahoo.com Abstract: In this study, a new Burr XII distribution is defined and studied. Various structural mathematical properties of the proposed model are investigated. The maximum likelihood method is used to estimate the model parameters. We assess the performance of the MLEs of the new distribution with respect to sample size n. The assessment was based on a simulation study. The new distribution is applied for modeling two real data sets to prove empirically its flexibility. The new Burr XII model can be viewed as a suitable model for fitting the right skewed and unimodal data. The new model provides adequate fits as compared to other Burr XII models by means of two applications.


Introduction
introduced another new system of frequency curves, analogously to the Pearson systemof densities, that includes twelve types of Cumulative Distribution Function (CDFs) which yield avariety of density shapes, this system is obtained by considering CDFs satisfying a differentialequation which has a solution, given by: where, ϒ(x) is chosen such that G(x) is a CDF on the real line and has twelve choices which madeby Burr, resulted in twelve models which might be useful for modeling data, the principal aim inchoosing one of these forms of distributions is to facilitate the mathematical analysis to which itis subjected, while attaining a reasonable approximation. A special attention has been devoted toone of these forms denoted by type XII whose CDF, G(x); is given as: both α and β are shape parameters, location and scale parameters can easily be introduced tomake (1) a fourparameter distribution. The corresponding Probability Density Function (PDF) isgiven by: The Burr XII Distribution (BXIID) originally proposed by Burr (1942), it has many applicationsin different areas. Coming early, Tadikamalla (1980) studied the BXIID and its related models.Some important extensions of the BXIID can be cited by Shao (2004), Zimmer et al. (1998), Soliman (2005), Wu et al. (2007), Silva et al. (2008), Silva et al. (2010a;2010b), Cordeiro et al. (2018), Afify et al. (2018), Altun et al. (2018a; and Yousof et al. (2018a;. (for more details about the BXIID see Burr (1942), (1968) and (1973), Burr and Cislak (1968), Hatke (1949) and Rodriguez (1977)). In this study, we propose a new BXII distributions, called the Burr-Hatke Exponentiated BXII Distribution (BHEBXII) by means of Burr-Hatke differential equation. In statistical literature, the so-called Burr-Hatke differential equation can be given asfollows: where, F = F(t) is the Cumulative Distribution Function (CDF) of a continuous random variableT and g(t,F) is an arbitrary positive function for any t 0 ∈ℜ. Equation (1) is considered bymany authors as a system of CDF(s) generator or simply a system of frequency curves. Using (1), Maniu and Voda (2008) introduced and studied the Burr-Hatke Distribution (BHD) with CDF and Probability Density Function (PDF) given by: The PDF corresponding to (2) is given by: Figure 1 displays some plots of the new density for some parameter values. Plots of the HRF of thenew model for selected parameter values are given in Fig. 2, where the HRF can be decreasing, increasing and unimodal.
We are motivated to introduce the BHEBXIID because it exhibits the decreasing, increasing and unimodal HRF as illustrated in Fig. 2. It is shown in Subsection 2.2 that the BHEBXIID can be viewed as a linear mixture of the BXII densities as illustrated in Equations (6) and (7). It can be viewed as a suitable model for fitting the unimodal and right skewed data as illustrated in Section 4. The BHEBXIID provide adequate fits as compared to other BXIIDs by means of two applications with small values for AIC, BIC, CAIC and HQIC. The proposed BHEBXIID is much better than the BXIID, Marshall. Olkin Burr XII (MOBXIID), TL Burr XII, Kumaraswamy Burr XII (KwBXIID), beta Burr XII (BBXIID), Beta Exponentiated Burr XII (BEBXIID), Five parameter beta Burr XII (FBBXIID), Five parameter Kumaraswamy Burr XII(FKwBXIID) and Zografos-Balakrishnan Burr XII (ZBBXIID) in modeling the breaking stress and the taxes revenue data.
The rest of the paper is outlined as follows. In section 2, we derive some statistical properties for the new model. Maximum likelihood estimation of the model parameters is addressed in section 3. Section 5 provides the simulation results. Two applications to real data sets to illustrate the importance of the new model are provided in section 5. Finally, we offer some concluding remarks in section 6. The PDF of the BHEBXIID The PDF of the BHEBXIID Properties Asymptotics Let a = inf{x|F θ,b,α,β (x) > 0} the asymptotics of CDF, PDF and HRF as x→a are given by: The asymptotics of CDF, PDF and HRF as x→∞ are given by: The effect of the parameters on tails of distribution can be evaluated by means of above equations.

Useful Expansions
In this section, mixture representations for Equations (2) and (3) are obtained. Consider the following expansions: and: Applying (4) for: in Equation (2) we get: where,b 0 = 1 and: Then, Equation (2) can be written as: and, for k≥ 1, we have: At the end, the CDF (2) can be written as: where,d 0 = 1-c k ,for k≥ 1 we have d 0 = -c k and: is the CDF of the EBXII. By differentiating (6), we obtain the same mixture representation: is the PDF of the EBXII. Using (4) the last expression can be rewritten as: where: is the BXII density with parameters α and β (1 + r) and: (7) reveals that the BHEBXII density function is a linear combination of the EBXIIdensity. Thus, some structural properties of the new family such as the ordinary and incompletemoments and generating function can be immediately obtained from wellestablished propertiesof the EBXIID.

Moments
The r th ordinary moment of X is given by: Then, we obtain: Setting n = 1, 2, 3 and 4 in (8), we have: The last results can be computed numerically for most parent distributions. The skewness andkurtosis measures can be calculated from the ordinary moments using well-known relationships.

Incomplete Moments
The n th incomplete moment of X is defined by: We can write from (7): where: are the beta and the incomplete beta functions of the second type respectively. Setting n = 1and 4 in (9), we have:

Moment Generating Function
The Moment Generating Function (MGF) of X, say M X (t) = E[exp (tX)], can be obtained from (7) as: ,..., | ,..., where, 1 i = − is the complex unit and L denotes an integration path (Gradshteyn and Ryzhik, 2000). The Meijer G-function contains as particular cases many integrals with elementary and special functions (Prudnikov et al., 1986). We now assume that α = m/β, where m and β are positive integers. This condition is not restrictive since every positive real number can beapproximated by a rational number. We have the following result, which holds for m and kpositive integers, µ>-1 and p> 0 (Prudnikov et al., 1992): We can write (for t< 0): Hence, the MGF of X can be expressed as:

Moment of Residual Life and Reversed Residual Life
The n th moment of the residual life, denoted by: The n th moment of the residual life of X is given by: ∫ Then, we can write:

Order Statistics
Order statistics make their appearance in many areas of statistical theory and practice. Suppose X 1:n , X 2:n ,..., X n:n , is a random sample from any BHEBXIID. Let X i:n denote the i th orderstatistic. The PDF of X i:n can be expressed as: We use the result 0.314 of Gradshteyn and Ryzhik (2000) for a power series raised to a positiveinteger n (for n≥ 1):

∑
We can demonstrate that the density function of the i th order statistic of any BHEBXIID can beexpressed as: where: denotes the EBXII density function with parameter (h + k + 1): Several approaches for parameter estimation were proposed in the literature but the maximum likelihood method is the most commonly employed. The Maximum Likelihood Estimators (MLEs) enjoy desirable properties and can be used for constructing confidence intervals and regions and in test statistics. The normal approximation for these estimators in large samples can beeasily handled either analytically or numerically. So, we consider the estimation of the unknownparameters of this model from complete samples only by maximum likelihood. Let x 1 ,…,x n be a random sample from the BHEBXII distribution with parameters θ,b, α and β. Let ϒ =(θ,b, α, β) T be the 4×1 parameter vector. For determining the MLE of ϒ, we have the log-likelihood function: The components of the score vector, L(ϒ) = , , , ,

Simulation Study
We simulate the new model by taking n=20, 50, 150, 300, 500 and 1000. For each sample size, weevaluate the ML Estimations (MLEs) of the parameters using the optim function of the R software.Then, we repeat this process 1000 times and compute the Averages of the Estimates (AEs) and Mean Squared Errors (MSEs). Table 1 gives all simulation results. The values in Table 1 indicatethat the MSEs of ˆ, , b θ α andβ decay toward zero when n increases for all settings ofθ,b,αandβ as expected under first-under asymptotic theory. The AEs of the parameters tend to be closerto the true parameter values when n increases. This fact supports that the asymptotic normaldistribution provides an adequate approximation to the finite sample distribution of the MLEs.
Data Set I: Breaking stress data. This data set consists of 100 observations of breaking stress of carbon fibres (in Gba) given by Nichols and Padgett (2006). Data Set II: Taxes revenue data.The actual taxes revenue data (in 1000 million Egyptian pounds).This data set were used by Nassar and Nada (2011) and Yousof et al. (2015) (see the data sets in Appendix B).
The Total Time Test (TTT) plot due to Aarset (1987) is an important graphical approach to verify whether the data can be applied to a specific distribution or not. According to Aarset (1987), theempirical version of the TTT plot is given by plotting: against r/n, where r = 1,...,n and y i:n (i = 1,...,n) are the order statistics of the sample. Aarset (1987) showed that the HRF is constant if the TTT plot is graphically presented as a straightdiagonal, the HRF is increasing (or decreasing) if the TTT plot is concave (or convex). The HRFis U-shaped (bathtub) if the TTT plot is firstly convex and then concave, if not, the HRF is unimodal. The TTT plots the three real data sets is presented in Fig. 3 and 4. This plotindicates that the empirical HRFs of the two data sets are increasing. In order to compare the fitted models, we consider the following goodness-of-fit statistics: The Akaike Information Criterion (AIC), Bayesian Information Criterion (BIC), Hannan-Quinn Information Criterion (HQIC), Consistent Akaike Information Criterion (CAIC), where: and: where, k is the number of parameters, n is the sample size, is the maximized log-likelihood. Generally, the smaller these statistics are, the better the fit.  Based on the values in Table 2-5 the BHEBXIID provides adequate fits as compared to BXIID, MOBXIID, TLBXIID, KwBXIID, BBXIID, BEBXIID, FBBXIID, FKwBXIID and ZBBXIID in application with small values for AIC, BIC, CAIC and HQIC. From our findings it is seen that in the applications cases considered here the proposed BHEBXIID turned out to be the best model in terms x Kaplan-Meier survival plot for data set II P-P plot for data set II of different selection criteria. Moreover, from the plots of estimated PDF against the observed histograms reveals that the new distribution provides closest fit to all the data sets. It may be mentioned that the new distribution has even outperformed the four and five parameter extensions considered the two applications. It is therefore is a useful contribution to the existing set of extended BXIID.

Conclusion
In this article, a new four parameter Burr-Hatke Exponentiated Burr XII Distribution (BHEBXIID) is defined and studied. Several structural mathematical properties of the proposed model areinvestigated. The Maximum Likelihood (ML) method is used to estimate the model parameters.We assess the performance of the MLEs of the new distribution with respect to sample size n.The assessment was based on a simulation study. The new distribution is applied for modelingtwo real data sets to prove its flexibility empirically. It is shown that the new lifetime model canbe viewed as a simple linear mixture of the Burr XII density. It can be viewed as a suitable model for fitting the unimodal and the right skewed data sets. The new model provides appropriate fitsas compared to other extensions of the Burr XII models by means of two real data applicationswith small values for AIC, BIC, CAIC and HQIC. Plots for the Estimated PDFs, P-P, TTT andKaplan-Meier Survival are provided for the two real data sets.
In this appendix we provide the densities used in the applications: