New Extended Burr Type X Distribution

This study introduces a new extended distribution of the Burr Type X distribution, using Marshall-Olkin method called the Marshall-Olkin Extended Burr Type X distribution. This study formulates the new distribution by inducing a tilt parameter in the Burr Type X distribution of Marshall-Olkin, to account for the significance of the size of an event. Model parameters are obtained using maximum likelihood and Bayesian methods of estimation. The scale and shape parameters are specifically defined to identify the dimensions and density of an event. Mathematical and statistical properties and limitations of the distribution are also presented. Lifetime data analysis is performed to demonstrate the model’s applicability and flexibility. Akaike and Bayesian Information Criteria illustrate that the new distribution provides better fit compared to other distributions,


Introduction
Twelve different forms of distributions were introduced by Burr (1942), where the Burr Type X and Burr Type XII have gained significant interest from many researchers. Among them, Rodriguez (1977) and Wingo (1993) who focused on the examination of the Burr Type XII distribution. Also, Al-Saiari et al. (2014) proposed Marshall-Olkin Burr Type XII distribution. This research article particularly discusses the properties of Burr Type X distribution and its associated relationship with other distributions, to build the ground for lifetime data analysis. The Burr Type X distribution fundamentally includes two parameters with the cumulative distribution function (CDF) as follows ( ; , ) = (1 − −( ) 2 ) ; > 0, , > 0 where λ and are scale and shape parameters, respectively (Raqab & Kundu 2006). Many researchers investigated the single-parameter model by, where λ=1. In this article, we will introduce the tilt parameter, which will result in a more flexible distribution compared to the original one.
For the single-parameter model, the probability density function (PDF) and CDF are written as: where, k is the shape parameter. Among the several studies that have been carried out on single-parameter Burr Type X distribution are; Ahmad Sartawi and Abu-Salih(1991), Jaheen (1995;1996), Ahmad et al. (1997), Raqab (1998), and Surles and Padgett (1998). The single parameter distribution of the extended Burr Type X by Surles and Padgett (2001) considered as a generalized Rayleigh distribution. This is characterized by normally distributed random variables with a zero mean and a constant variance, and has no correlation, and therefore, making it more applicable, instead of restricting its applicability for lifetime data. For instance, this model can be applied to estimate the magnitude of noise variance in the Magnetic Resonance Imaging (Sijbers et al. 1999 New extended families of distribution were proposed with the inclusion of an additional parameter using Marshall and Olkin method which provide more flexible distribution and represented a wider range of behaviours (Marshall & Olkin 1997). This paper intends to study the properties of the Marshall Olkin extended Burr Type X (MOEBX) distribution. The paper first illustrates the construction of MOEBX in Section 2. Section 3 provides graphical illustrations of the dimensions of MOEBX distribution. Parameter estimation using Maximum Likelihood and Bayesian methods are presented, respectively, in Sections 4 and 5. Section 6 focuses on defining the statistical properties of MOEBX distribution along with application to real lifetime data. Finally, some concluding remarks are given in Section 7.

Construction of the Extended Burr Type X Distribution of Marshall-Olkin
To construct the survival function of the proposed MOEBX distribution, ̅ ( ; ), we need to obtain the survival function, ̅ ( ) = 1 − ( ) , of the Burr Type X distribution. This can be given by: where, is the tilt parameter, which measures the shape of the rising boundaries (Jose 2011). The CDF and PDF for MOEBX distribution are respectively written as: and where > 0, > 0.

Properties of the Extended Burr Type X Distribution of Marshall-Olkin
Assume a random variable X that follows the MOEBX distribution, denoted by X ~ MOEBX (α, k). Figure 1 below depicts the cumulative and probability density functions of MOEBX distribution (α, k) at different values of its parameters, α and k. From Figure 1, it is clear that the PDF takes various shapes, depending on the values of α and k: i. For k > 0.5 at varying values of α, the density function is unimodal, i.e. g(x; α, k) is skewed towards right ii.

The Hazard Function
The hazard rate of the original Burr Type X distribution, h(x) is directly related to the hazard rate of the extended distribution of Marshall-Olkin r(x; α), which is given by: In which, h(x) is the corresponding hazard rate of f(x).From the above equation (4), Therefore, the hazard rate function and the survival function of the MOEBX distribution are respectively given by ii. For at varying values of α, r(x; α, k) decreases after reaching its maximum, which can be clearly observed when α ≤ 1. Moreover, the relationship depends on the range of the new parameters α > 1 or 0 < α < 1, (Marshall & Olkin 1997). Thus, for 0 < α < 1 > 0, it can be shown that: While for α > 1, it follows that:

The Quantile, Median, and Mode
The qth quantile of the MOEBX distribution is: where −1 ( )represents the inverse function of the MOEBX distribution. Then replacing q with 0.5, the median of the MOEBX distribution is: Solving the equation ( ; , ) , the mode of MOEBX distribution can be found by: = 0 (13) Table 1 reports the modes for some given values of k and α, which illustrates that the PDF of MOEBX distribution is unimodal for these values of the parameters.

Maximum Likelihood Estimation
Maximum likelihood (ML) method is used to obtain the parameters of the MOEBX distribution. Assuming that we have a random sample = ( 1 , 2 , … . ) of size n from MOEBX distribution, the log-likelihood of MOEBX is given by:

Approximate Confidence Intervals:
It is clear that we cannot obtain the ML estimates of the parameters of MOBEX distribution in close form. For this reason, approximations are used to compute the confidence interval of ̅ , the estimates of which are rooted in the asymptotic distribution of ML approach. This is based on the following assumptions:  large size of the sample is taken for the estimations of ̂̅ ,  Estimates follow a normal distribution i.e. Then using this approach, the confidence interval of the parameters at the level of 100 (1 − 2 ) % will be approximated as: where 2 ⁄ is the standard normal distribution with 2 ⁄ th percentile.
In order to evaluate whether the MOEBX distribution fits better on the data compared to the Burr Type X distribution, the Likelihood ratio (LR) statistical test is conducted. That is, the null hypothesis 0 : = 1 is tested against 1 : ≠ 1 and the test statistic of is computed by:

Bayesian Estimation
Suppose that for a random sample = ( 1 , 2 , … . ) of size n from MOEBX distributionwith the unknown parameters ̅ = ( , ) which are assumed to be independent having non-informative prior distributions such that α ~uniform (a 1 ,b 1 ) and k ~uniform (a 2 ,b 2 ). Then the joint prior for and can be written by Therefore, the subsequent joint posterior distribution can be written as where 3 ( ), 3 ( ; , )are as defined in (15), and is the normalizing constant, which ensures that the area represents the probability density function. Markov Chain Monte Carlo technique can be used to obtain the posterior estimates of the parameters for MOEBX under squared error loss function.

Application to Real Life Data
Lawless (2011) described the breaking strength T of single carbon fibers of different lengths which consisted of 70 observations. The Kolmogorov-Smirnov (K-S) statistic and the empirical CDF were used to validate the fitted model. The empirical CDF and the fitted CDF is shown in Figure 3 by replacing the parameters with their ML estimates. The K-S test statistic is 0.1571 with the p-value = 0.3531 using ML estimates. When using Bayesian estimates the K-S test value is 0.2 with the p-value = 0.1216. Therefore, both K-S tests values and Figure 3 indicate that MOEBX distribution provides the appropriate fit for this data set. The ML and Bayesian estimates of the MOEBX parameters, standard error (SE), 95% confidence intervals (CI), and interval length are illustrated in Table 2. To examine the flexibility of the proposed MOEBX distribution, the same set of real data has been used to fit the model, using ML and Bayesian methods of estimation. ML and Bayesian estimates, Akaike information criterion (AIC) and Bayesian information criterion (BIC) are reported in Table 3 in order to compare the two distributions. This was conducted using Algorithms 1(see Appendix).
The LR test statistics for testing the hypotheses 0 : = 1 versus 1 : ≠ 1 is Λ=146.33 (p-value < 0.05) which indicates that the MOEBX distribution is more suitable than Burr Type X distribution for this particular data set. The results indicate that the MOEBX distribution, in comparison with the Burr X distribution, has by far the lowest statistics of AIC and BIC values. Thus, it can be asserted that MOEBX distribution is more flexible than the Burr Type X distribution. Moreover, the normal quantile-quantile plot shows that the error of the fitted MOEBX distribution tends to approximate normality, (See Figure 4).

Conclusion
This paper aimed to propose a MOEBX distribution using Marshall-Olkin method by adding a tilt parameter as an extension to the Burr Type X distribution. The mathematical and statistical properties of the MOEBX distribution are discussed and its parameters are estimated using ML and Bayesian methods. The confidence interval for the parameters is asymptotic and based on certain assumptions regarding the mean, variance and variancecovariance matrix. A realife data set was fitted to the proposed distribution to illustrate the flexibility of the MOEBX distribution. Parameter estimates were obtained using ML and Bayesian methods. The standard errors of the estimates and lengths of the intervals showed that the Bayesian estimates provide a better fit for the data than the ML estimates.