The Three-parameters Marshall-Olkin Generalized Weibull Model with Properties and Different Applications to Real Data Sets

A new three-parameter life parametric model called the Marshall-Olkin generalized Weibull is defined and studied. Relevant properties are mathematically derived and analyzed. The new density exhibits various important symmetric and asymmetric shapes with different useful kurtosis. The new failure rate can be “constant”, “upside down-constant (reversed U-HRF-constant)”, “increasing then constant”, “monotonically increasing”, “J-HRF” and “monotonically decreasing”. The method of maximum likelihood is employed to estimate the unknown parameters. A graphical simulation is performed to assess the performance of the maximum likelihood estimation. We checked and proved empirically the importance, applicability and flexibility of the new Weibull model in modeling various symmetric and asymmetric types of data. The new distribution has a high ability to model different symmetric and asymmetric types of data.


Introduction
Consider a baseline reliability function (RF) of the Weibull (W) distribution (Weibull (1951)) with probability density function (PDF) (4) In this paper, we propose and study a new generated Weibull model called the Marshall-Olkin generalized Weibull (MOGW) distribution and give a comprehensive description of its mathematical properties. In fact, the MOGW model is motivated by its importance flexibility in application. By means of two applications, it is noted that the MOGW model provides better fits than other models each having the same number of parameters.
where and 2 are two positive shape parameters representing the different patterns of the MOGW distribution. The corresponding PDF of (5) is given by Henceforth, ∼ MOGW( , 1 , 2 ) denotes a random variable having density function (6). The MOGW distribution is motivated by the following motivations. Suppose a system is made up of independent components in series, where is a random variable with geometric distribution and probability mass function Pr( = ) =̇− 1 , = 1,2, . . . and ∈ (0,1). Suppose that random variables 1 , 2 , . .. , represent the lifetimes of each component and suppose that they have the generalized W distribution. Then a random variable = ( 1 , 2 , … , ) represents the time to the first failure with CDF (5).Form another view, consider now a parallel system with independent components and suppose that a random variable has geometric distribution with the probability mass function ( = ) = −1̇−1 , = 1,2, . . . and > 1. Let 1 , 2 , . .. as before representing the lifetimes of each component and suppose that they have the generalized W distribution. Then a random variable = ( 1 , 2 , . . . , ), represents the lifetime of the system. Therefore, the random variable follows (5). The reliability function (rf), hazard rate function (HRF) and cumulative hazard rate function (cHRF) of are, respectively, given by ( ) = The MOGW distribution includes the generalized Weibull (GW) distribution when = 1 . For 2 = 1 , we obtain the MO Weibull (MOW) model. For 1 = 1 , we have the MOG-exponential (MOGE) distribution. For 1 = 2 , we obtain the MOG-Rayleigh (MOGR) distribution. Figure 1 gives some plots of the MOGW PDF (left) and HRF (right). From Figure 1 (left) we conclude that the PDF MOGW distribution have various symmetric and asymmetric shapes with different kurtosis. From Figure  1 (right) we note that the HRF MOGW model can be "constant", "upside down-constant (reversed U-HRF)", "increasing then constant", "monotonically increasing", "J-HRF" and "monotonically decreasing".    We also conclude that the proposed model is much better than the Odd Lindley exponentiated Weibull, gamma exponentiated-exponential, odd Weibull Weibull models, and a good alternative to these models in modeling survival times of Guinea pigs. Finally, the proposed model is much better than the Odd Lindley exponentiated Weibull, exponentiated Weibull, transmuted Weibull, odd Log Logistic Weibull models, and a good alternative to these models in modeling glass fibers data.

Mathematical properties
where where 0 = and =(−1)̇( 2 ) using (7) and (8) the CDF of the MOGW model in (5) can be expressed as , the PDF of the MOGW model can also be expressed as a mixture of expW densities. By differentiating , 1 , 2 ( ) , we obtain the same mixture representation where ( ) is the exp W PDF with power parameter ( ) . Equation (9) reveals that the MOGW PDF is a linear combination of exp W PDFs. Thus, some structural properties of the new family such as the ordinary and incomplete moments and generating function can be immediately obtained from well-established properties of the exp W distributions.

Incomplete moments
The main applications of the first incomplete moment refer to the mean deviations and the Bonferroni and Lorenz curves. These curves are very useful in economics, reliability, demography, insurance and medicine. The ℎ incomplete moment, say ( ) , of can be expressed from (9) as

Probability weighted moments
The PWM method can generally be used for estimating parameters of a distribution whose inverse form cannot be expressed explicitly. The ( , ) ℎ PWM of following the MOGW distribution, say , , is formally defined by Using equations (5) and (6), we can write , 1 , 2 ( ) , 1 , 2 ( ) = ∑∞ =0̇( ) where Then, the ( , ) ℎ PWM of can be expressed as , = (  Setting the nonlinear system of equations = 2 = and 1 = 0 and solving them simultaneously yields the MLEs. To solve these equations, it is usually more convenient to use nonlinear optimization methods such as the quasi-Newton algorithm to numerically maximize ℓ . For interval estimation of the parameters, we obtain the 3 × 3 observed information matrix ( ) = { 2 ℓ } (for , = , ζ 1 , 2 )whose elements can be computed numerically.

Graphical assessment
We perform a graphical simulation in order to assess of the finite sample behavior of the MLEs. The assessment was based on the following algorithm: I-Using the quantile function, we generate 1000 samples of size from the MOGW distribution and compute the MLEs for the 1000 samples. II -Compute the SEs of the MLEs for the 1000 samples. III -Compute the biases and mean squared errors given for all parameters. We repeated these steps for = 50, 100, … ,250, so computing biases, mean squared errors (MSEs) for , ζ 1 , 2 . Figure 2 shows how the three biases vary with respect to . Figure 3 shows how the three MSEs vary with respect to . From Figure 2 and 3, the biases for each parameter are generally "negative" and decrease to zero as → ∞ , the MSEs for each parameter decrease to zero as → ∞.

Applications
In this section, we provide four real applications to show empirically its potentiality. In order to compare the fits of the MOGW distribution with other competing distributions, we consider (statistic) and the Anderson-Darling (statistic) . The MLEs and its standard errors (SEs) are given in Table 1, Table 3, Table 5 and Table 7. The values of (statistic) and (statistic) are listed in Table 2, Table 4, Table 6 and Table 8. The total time in test (TTT), probability-probability (P-P) plots, Estimated PDF (EPHF), EHRF for data sets I, II, III and IV of the proposed models are displayed in Figure  4, Figure 5, Figure 6

Modeling failure times data
The data consist of 84 observations. This data is recently analyzed by (Khalil et al. (2019) and Mansour et al. (2010b, c). In Table 1 and Table 2 Figure 4: TTT plot, P-P plot, EPHF, EHRF for failure times data. 6.2 Modeling cancer data This data set represents the remission times (in months) of a random sample of 128 bladder cancer patients as reported in Lee and Wang (2003). This data is recently analyzed by (Khalil et al. (2019) and Mansour et al. (2010b, c).We compare the fits of the MOGW distribution with other competitive models, namely: The TMW, MBW, transmuted additive W distribution (TAW) (Elbatal and Aryal, (2013)), exponentiated transmuted generalized Rayleigh (ETGR) (Afify et al. (2015)), and the Weibull distributions with corresponding densities (for > 0 ). Based on the figures in Table 4 we conclude that the proposed MOGW lifetime model is much better than the W, TMW, MBW, TAW, ETG-R models with (statistic) = 0.0672and (statistic) = 0.4214.   Figure 5: TTT plot, P-P plot, EPHF, EHRF for remission data set.

Modeling survival times
The second real data set corresponds to the survival times (in days) of 72 guinea pigs infected with virulent tubercle bacilli (see Bjerkedal (1960)). This data is recently analyzed by (Khalil et (2012)). Based on the figures in Table 6 we conclude that the proposed MOGW model is much better than all other models with (statistic) = 0.0961 and (statistic) = 0.6897. Figure 6: TTT plot, P-P plot, EPHF, EHRF for survival data.

Concluding remarks
This article presented a new three-parameter life parametric model called the Marshall-Olkin generalized Weibull (MOGW) model. Some of its relevant structural properties are derived and analyzed. The new density is expressed as a linear mixture of the exponentiated Weibull density. The density of the MOGW distribution exhibits various important symmetric and asymmetric shapes with different useful kurtosis. The HRF of the MOGW model can be "constant", "upside down-constant (reversed U-HRF-constant)", "increasing then constant", "monotonically increasing", "J-HRF" and "monotonically decreasing". The maximum likelihood method is employed to estimate the unknown model parameters. A graphical simulation is performed to assess the performance of the maximum likelihood estimation biases, mean squared errors. It is noted that the three biases are generally "negative" and decrease to zero as → ∞ , the mean squared errors for each parameter decrease to zero as → ∞. We checked and proved empirically the importance, applicability and flexibility of the new Weibull model in modeling various symmetric and asymmetric types of data. The new distribution has a high ability to model different symmetric and asymmetric types of data.