The Extended Marshall-Olkin Burr III Distribution: Properties and Applications

We study a new continuous distribution called the Marshall-Olkin modified Burr III distribution. The density function of the proposed model can be expressed as a mixture of modified Burr III densities. A comprehensive account of its mathematical properties is derived. The model parameters are estimated by the method of maximum likelihood. The usefulness of the derived model is illustrated over other distributions using a real data set.


Introduction
Recently, several lifetime models have been derived and used in modeling data in several areas. In 1942, Burr proposed a system of twelve types of distribution functions based on generating the Pearson differential equation (Burr, 1942). Among these Burr distributions, the Burr III model which is extensively used to model data in several fields. For example, forestry data (Gove et al., 2008 andLindsay et al., 1996), fracture roughness data Kotz, 2006a and, life testing (Wingo, 1993), meteorology (Mielke, 1973), modeling crop rice (Tejeda and Goodwin, 2008) and reliability data (Abdel-Ghaly et al., 1997). The probability density function (pdf) of the MBIII distribution is defined (for > 0) by where > 0 is a scale parameter and > 0 and > 0 are shape parameters.
The cumulative distribution function (cdf) and hazard rate function (hrf) of the MBIII distribution are given by In this paper, we propose and study a new four-parameter model called the Marshall-Olkin modified Burr III (MOMBIII) distribution. In fact, we constrcut the new model based on the Marshall-Olkin-G (MO-G) family proposed by Marshall and Olkin (1997). Further, we provide an account of its mathematical properties.
The MO-G family (Marshall and Olkin, 1997) has been used extensively to generalize many well-known distributions. For example: the MO exponential and MO Weibull due to Marshall and Olkin (1997) Consider the baseline cdf, ( ), then the survival function (sf) of MO-G family is where ̅ ( ) = 1 − ( ) is the baseline sf and ̅ = 1 − , > 0 is a shape parameter. For = 1, we obtain the baseline distribution.
The pdf of MO-G family reduces to The hrf is given by The rest of the paper is outlined as follows: In Section 2, we define the MOMBIII distribution and provide some plots for its pdf and hrf. Some mathematical properties including linear representation for its pdf, ordinary moments, order statistics, Rényi entropy and probability weighted moments (PWMs) are calculated in Section 3. We discuss the maximum likelihood estimation of the model parameters in Section 4. In Section 5, we assess the performance of the maximum likelihood estimates via a simulation study. Using a real data set, we show the importance of the new model in Section 6. Finally, some concluding remarks are given in Section 7.

The MOMBIII distribution
The cdf of the MOMBIII distribution is given (for > 0) by The corresponding pdf comes out as where , and are positive shape parameters, and is a positive scale parameter.
The hrf of the MOMBIII reduces to The quantile function of the MOMBIII is given by  (6) for some selected parameter values. The figure shows that the shape of the density function is flexiable from reversed J-shape to concave down shape for certain parameter values. The plots of the MOMBIII hrf are displayed in Figure 2. The MOMBIII allows for great flexibility and hence it can be very useful in many practical situations for modeling positive data. Properties of the MOMBIII distribution.

Useful expansion
In this section, we provide a useful linear representation for the pdf of the MOMBIII distribution. The pdf (6) can be expressed as  For | | < 1, the general binomial series holds Using (8) in (7), we have The last equation can be rewritten as where = (1 − −1 ) / and ( +1) ( ) is the density function of the MBIII with shape parameters ( + 1) and and a scale parameter . Based on Equation (9), we can derive the properties of the MOMBIII distribution from those of the MBIII distribution.

Moments
Let be a random variable having the MBIII distribution, then the th ordinary moments of is given (for < ) by is the beta function of the second kind. Further information abiut the MBIII distribution can be found in .
The th moment of follows from (9) (for < ) as The moment generating function (mgf) of the MOMBIII distribution is given by  Figure 4 displays the plots of skewness and kurtosis of the MOMBIII distribution in terms of and when = 5 and = 4. From Figure 4 and the corresponding data values (not included), the skewness is always positive which indicates that the MOMBIII distribution is right skewed, and the kurtosis is increasing function for and .

Order statistics
Let 1 , 2 , … … , be a random samples of size from the MOMBIII distribution and its ordered values are (1) , (2) , … … , ( ) . Then, the pdf of the th order statistic, say : , can be written as Using (5) and (6), we have Applying the generalized binomial series to the dominator, we obtain Substituting in (12), we can write Combining the last equation with equation (11), we have

Rényi entropy
The Rényi entropy of the rv is a measure of variation of the uncertainty and it is defined (for > 0 and ≠ 1) by . Using (6), we can write Using the generalized binomial series and after some simplifications, we obtain Then, the Rényi entropy of X can be expressed as ].

Probability weighted moments
The ( , )th PWM of is defined by Using equation (12) Using the generalized binomial series, the above equation reduces to Or equivalently, we have

Maximum likelihood estimation
In this section, the maximum likelihood estimators (MLEs) of the MOMBIII parameters are obtained. Let 1 , ⋯ , be a random sample of size from this distribution with parameters , , and . Let =( , , , ) ú be the × 1 parameter vector. The log-likelihood function for reduces to ℓ( ) = log + log + log − ( + 1) ∑ log(1 + − ) The above log-likelihood can be maximized numerically using the R (optim function), SAS (PROC NLMIXED) or Ox program (sub-routine MaxBFGS)), among others.
The score vector elements are given respectively by The exact solution of above derived MLEs for the unknown parameters is not possible. So it is more convenient to use nonlinear optimization algorithms such as Newton Raphson algorithm to numerically maximize the above likelihood function. For interval estimation of the parameters, we obtain the × observed information matrix ( ) = { ∂ 2 ℓ ∂ ∂ } (for , = , , , ), whose elements can be obtained upon request.

Simulation study
In this section, we assess the finite sample behaviors of the MLEs for the MOMBIII distribution via a Monte Carlos simulation study. We generate 10,000 samples of sizes = 50 and = 300 for different combinations of parameters. The MATHEMATICA 10.0 is used to obtain the mean values, bias and mean square errors (MSE). Table 1 lists the mean values of the estimates, bias and MSE.
One can see, from Table 1, that the estimates of model parameters are closer to true values as sample size increases and the biases and MSE decreases as → ∞.

Data analysis
In this section, we illustrate the importance and flexibility of the MOMBIII distribution using a real data set. we compare the fits of the MOMBIII model with the MBIII, beta exponential (BE) ( where the pdf of the MBIII model is given in Section 1 and all the parameters are real numbers except for the TMBIII and TG distributions for them | | ≤ 1. We consider a data set obtained from Smith and Naylor (1987) which represents the strengths of 1.5 cm glass fibers, measured at the National Physical Laboratory, England.  9.0960. The descriptive summary of the data set is given in Table 2.     The distribution parameters are estimated by using the maximum likelihood method. The estimated values are given in Table 3. In order to compare the fitted distribution, we used the following goodness-of-fit statistics: -2 log-likelihood (−2 ), Akaike information criterion ( ), corrected Akaike information criterion ( ), Bayesian information criterion ( ) and Hannan-Quinn Information Criterion ( ). Along with these measures some most commonly used statistics are also used, such as: Durbin-Watson test , Anderson Darling test and Kolmogorov-Smirnov (K-S) test statistic with its corresponding p-value. Table 3 shows the goodness-of-fit statistic results.
It can be concluded that the MOMBIII distribution has the lowest −2 , , , , , , and the K-S statistic values, and the largest p-values among all the other models, hence, the MOMBIII distribution could be chosen as the best model. The relative histogram and the fitted distributions is displayed in Figure 5. Also, the plots of the fitted cdfs and empirical cdf of the data are displayed in Figure 6.

Conclusion
In this article, we propose a new four-parameter lifetime model called the Marshall-Olkin modified Burr III (MOMBIII) distribution which extends the modified Burr III (MBIII) model. The density function of the MOMBIII model can be expressed as a mixture of MBIII densities. We provide some mathematical properties of the MOMBIII distribution including explicit expressions for the ordinary moments, order statistics, Rényi entropy and probability weighted moments. The unkown parameters of the new model are estimated by maximum-likelihood and a Monte Carlo simulation study to assess the finite sample behavior of the maximum-likelihood estimators is performed. By means of an application to real data, we prove empirically that the MOMBIII distribution can give better fits than other well-known models in the literature.