Marshall-Olkin Zubair-G Family of Distributions

A new class of distributions called Marshall-Olkin Zubair-G family is proposed in this study. Some statistical properties of the family are derived and two special distributions namely, Marshall-Olkin Zubair NadarajahHaghighi and Marshall-Olkin Zubair Weibull distributions are developed. The plots of the density and hazard rate functions of the special distributions exhibit different shapes for chosen parameter values, making them good candidates for modeling different types of datasets. A real life application using the Marshall-Olkin Zubair Nadarajah-Haghighi distribution revealed that it performs better than other existing extensions of the NadarajahHaghighi distribution for the given dataset.


Introduction
The quest to develop flexible probability distributions have become an issue of interest to myriad of researchers owing to the usefulness of these distributions in modeling datasets and making inference in areas such as engineering, financial and biological modeling among others. This recent development has led to the proposition of several generalized classes of distributions called generators in literature for modifying existing distributions. The generators usually add one or more extra parameter (s) to the existing classical distributions to make them capable of modeling datasets that exhibit different traits such as bimodality, heavy-tail, monotonic and non-monotonic failure rates, symmetric and non-symmetric shapes. Hence, the aim is to develop new distributions that provide reasonable parametric fit to datasets obtained from different fields.
However, it is worth noting that no single probability distribution can provide good fit to all kinds of datasets. Thus, there is the need to develop new probability distributions for modeling datasets. Some common generators that have been developed for modifying existing distributions include: Marshall-Olkin alpha power family (Nassar et al., Marshall Gx is the CDF of the baseline distribution. The objective of this study is to develop another extension of the Zubair-G family called Marshall-Olkin Zubair (MOZ)-G family by adding an extra shape parameter to the Zubair-G family to make it more flexible. The Zubair-G family adds only a single scale parameter 0   to the baseline distribution. Thus, if the baseline distribution has no shape parameter as in the case of the exponential distribution, the resulting distribution will lack shape parameter. But to produce distribution with heavy-tail, and control skewness and kurtosis, a shape parameter is required. It is therefore necessary to add an extra shape parameter to the Zubair-G family.
Suppose () Zx is a baseline CDF which depends on a parameter vector 12 ( , ,..., ) T Then the CDF of the Marshall-Olkin family is defined as where  is an extra shape parameter. Substituting equation (1) into equation (2), the CDF of the MOZ-G family of distribution is defined as 2 2 ( ; ) The probability density function (PDF) related to the MOZ-G family is given by Hence, the CDF of the MOZ-G family is obtained. Alternatively, the CDF in equation (3) can be interpreted as follows: suppose the random variable For the sake of simplicity, ( ; ) Gx can be written as () Gx and a random variable X that follows the MOZ-G family is represented by The MOZ-G family has a tractable CDF making it easy to obtain random observations from the family provided the CDF of the baseline distribution is also tractable. The quantile function of the MOZ-G family is given by The mixture representation of the density function is useful when deriving the structural properties of the MOZ-G family of distributions.

Statistical Properties
This section presents some useful statistical properties of the MOZ-G family of distributions. The statistical properties derived are the moments, moment generating function (MGF), entropies, stochastic ordering and order statistics.

Moments and Moment Generating Function
The th r non-central moment can be expressed in terms of the quantile function of the baseline distribution. Letting () G x u = , the th r non-central moment can be expressed as Qu is the quantile function of the baseline distribution with CDF () Gx. The MGF of a random variable X that follows the MOZ-G family of distributions if it exists is given by Alternatively, the MGF can be expressed in terms of the quantile function of the baseline distribution as

Entropy Measures
Entropies are measures of variation of a random variable. This section presents the Rényi and  − entropies. The Rényi entropy (Rényi, 1961) of a random variable X with density function () fx is given by Also, 1 X is said to be stochastically greater than 2 X in the i. hazard rate order If 1 X is MOZ-G random variable and 2 X is Zubair-G random variable, then the likelihood ratio is ( )

Order Statistics
Order statistics are useful in quality control and reliability analysis. This section presents the density function of the th p order statistic and its th r non-central moment. Let 1:

Parameter Estimation
The maximum likelihood technique is employed to estimate the parameters of the MOZ-G family of distributions.
Suppose that 12 To obtain the estimators for the parameters, the score functions are equated to zero and the resulting system of equations are solved numerically. In order to find the interval estimates of the parameters, the observed information matrix can be computed as likelihood ratio (LR) test is carried out using the following hypotheses: 0

Special Distributions
In this section, two special cases of the MOZ-G family of distributions are discussed.

Marshall-Olkin Zubair Nadarajah-Haghighi (MOZNH) Distribution
Given that the baseline CDF is that of the Nadarajah-Haghighi (NH) distribution. That is, and the corresponding density function is The PDF of the MOZNH distribution is given by  The hazard rate function of the MOZNH distribution is given by The hazard rate function for the MOZNH distribution can exhibit non-monotonic failure rates such as the upsidedown bathtub for some given parameter values.

Simulation Studies
This section presents the Monte Carlo simulation results used to assess the performance of the estimators of the parameters. For illustration purpose the MOZNH distribution was used for the simulation. The experiment was repeated 10, 000 times with sample sizes 25,50, 75,100 n = and 125 . The root mean square error (RMSE) for the parameters as shown in Table 1 decays to zero as the sample size increases. The coverage probabilities (CP) for the 95% confidence interval for the parameters in some cases were quite close to the nominal level of 0.95 .
Marshall-Olkin Zubair-G Family of Distributions 205

Empirical illustration
Here, we demonstrated the application of the MOZNH distribution using dataset. The dataset comprises the remission time of 128 bladder cancer patients presented in Lee and Wang (2003)  We compared the performance of the MOZNH distribution with that of the Zubair NH (ZNH), exponentiated NH (ENH) (Abdul-Moniem, 2015) and Kumaraswamy NH (KNH) (Lima, 2015) distributions using the Akaike information criterion (AIC), consistent Akaike information criterion (CAIC) and Bayesian information criterion (BIC). The Kolmogorov-Smirnov (K-S), Cramér-von Mises (CM) and Anderson-Darling (AD) statistics were used to investigate the goodness-of -fit of the models. The density functions of the ENH and KNH distributions are respectively given by:  Table 2 presents the maximum likelihood estimates of parameters of the fitted distributions with their corresponding standard errors and 95% confidence intervals (CI). The maximum likelihood estimates of the parameters were obtained by maximizing the log-likelihood functions of the fitted distributions via the subroutine mle2 using the bbmle package in R (Bolker, 2014). The optimizations were carried out using the BFGS technique and the initial values for the optimization were obtained using the GenSA package in R. The estimates of the parameters were all significant at the 5% level.  Table 3 displays the model selection criteria and the goodness-of-fit statistics for investigating how well the distributions fit the given dataset. The results indicate that the MOZNH distribution provides a better fit to the datasets than the other candidate distributions because it has the least values for thse model selection criteria and the goodness-of-fit statistics. The LR test was conducted to compare the performance of the MOZNH distribution with the ZNH distribution. The test yielded a test statistic of 12.6260 with corresponding p-value of 0.0004. This is an indication that the MOZNH distribution gives a better fit to the dataset than the ZNH distribution. Figure 5 displays the density plots and the distribution function plots of the fitted distributions. The plots revealed that the MOZNH distribution fits the dataset well.
Marshall-Olkin Zubair-G Family of Distributions 207 The probability-probability plots of the MOZNH, ZNH, ENH and KNH distributions for the dataset are presented in Figure 6. Figure 6 revealed that the MOZNH distribution fitted the dataset well.

Conclusion
The MOZ-G family of distributions was developed in this study. The proposed generator was used to develop the MOZNH and MOZW distributions. The density and hazard rate functions of the MOZNH and MOZW distributions exhibit different type of shapes making them suitable for analyzing datasets with either monotonic or non-monotonic failure rates. The application of the MOZNH distribution was illustrated using datasets and the findings indicated that the distribution fitted the dataset well.