The four-parameter exponentiated Weibull model with Copula, properties and real data modeling

A new four-parameter lifetime model is introduced and studied. The new model derives its flexibility and wide applicability from the well-known exponentiated Weibull model. Many bivariate and the multivariate type versions are derived using the Morgenstern family and Clayton copula. The new density can exhibit many important shapes with different skewness and kurtosis which can be unimodal and bimodal. The new hazard rate can be decreasing, J-shape, U-shape, constant, increasing, upside down and increasing-constant hazard rates. Various of its structural mathematical properties are derived. Graphical simulations are used in assessing the performance of the estimation method. We proved empirically the importance and flexibility of the new model in modeling various types of data such as failure times, remission times, survival times and strengths data.

The four-parameter exponentiated Weibull model with Copula, properties, and real data modeling 651 (a) (b) Figure 1: Plots of the MOLEW PDF. Reduced model

Properties Moments
First the quantity [1 − (1 − − 1 ) 2 ] can expanded as where 0 = 2 and and then where 0 = and using (7) and (8) the CDF of the MOL-G family in (5) can be expressed as the PDF of the MOLEW model can also be expressed as a mixture of exponentiated W (EW) PDF. By differentiating ( ) , we obtain the same mixture representation where * = 1 + 1 and * , 2 , 1 ( ) is the EW PDF with power parameter ( * ) . Equation (9) reveals that the MOLEW density function is a linear combination of EW densities. Thus, some structural properties of the new family such as the ordinary and incomplete moments and generating function can be immediately obtained from wellestablished properties of the EW distribution. The r th ordinary moment of is given by then we obtain and The last integration can be computed numerically for most parent distributions. The skewness and kurtosis measures can be calculated from the ordinary moments using well-known relationships. The moment generating function (MGF) ( ) = ( ) of . Clearly, the first one can be derived from equation (9) as The s th incomplete moment, say ( ) , of can be expressed from (9) as setting = 1,2,3,4 in (11) we get

Order statistics
Suppose 1 , … , is any random sample from any MOLEW distribution. Let ℏ: denote the i th order statistic. The PDF of ℏ: can be expressed as Following Nadarajah et al. (2015), we have

Simple type Copula based construction
In this Section, we consider several approaches to construct the bivariate and the multivariate MOLEW type distributions via copula (or with straightforward bivariate CDFs form, in which we just need to consider two different then we have a 9-dimension parameter model.

Via Clayton copula The bivariate extension
The bivariate extension via Clayton copula can be considered as a weighted version of the Clayton copula, which is of the form 1 + 2 . This is indeed a valid copula. Next, let us assume that ∼ MOLEW ( 1 , 1 , 2 , 1 ) where and ∼ MOLEW ( 1 , 1 , 2 , 1 ) where the associated CDF bivariate MOLEW type distribution will be

The Multivariate extension
A straightforward -dimensional extension IRom the above will be Further future works could be be allocated for studying the bivariate and the multivariate extensions of the MOLEW model.

Estimation
Let 1 , … , be a random sample from the MOLEW distribution with parameters , , 2 and 1 . Let = ( , , 2 , 1 ) be the 4 × 1 parameter vector. For determining the MLE of , we have the log-likelihood function ℓ = ℓ( ) = + The components of the score vector are easily to be derived.

Graphical assessment
Graphically, we can perform the simulation experiments to assess of the finite sample behavior of the MLEs. The assessment was based on the following algorithm:

Applications
In this section, we provide four applications of the OLEW distribution to show empirically its potentiality. In order to compare the fits of the MOLEW distribution with other competing distributions, we consider the Cramér-von Mises CVM and the Anderson-Darling (AD). These two statistics are widely used to determine how closely a specific CDF fits the empirical distribution of a given data set. These statistics are given by

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 given in Appendix (b). We compare the fits of the MOLEW distribution with other competitive models, namely: The TMW, MBW, transmuted additive W distribution (TA-W) (Elbatal and Aryal, (2013)), exponentiated transmuted generalized Rayleigh (ETGR) ), and the W (W, (1951)) distributions with corresponding densities (for > 0 ). Based on the figures in Table 5 we conclude that the proposed MOLEW lifetime model is much better than the W, TM-W, MB-W, TA-W, ETG-R models with small values for CVM and A.D in modeling cancer patient's data.

Modeling survival times
The second real data set corresponds to the survival times (in days) of 72 guinea pigs infected with virulent tubercle bacilli reported by Bjerkedal (1960). This data is given in Appendix (c). We shall compare the fits of the MOLEW distribution with those of other competitive models, namely: Odd Lindley exponentiated W (OLEW), the Odd W-W (OW-W) (Bourguignon et al., (2014)), the gamma exponentiated-exponential (GaE-E) (Ristic and Balakrishnan (2012)) distributions, whose PDFs (for > 0 ). Based on the figures in Table 7 we conclude that the proposed MOLEW model is much better than the OLEW, OW-W, WLog-W, GaE-E models, and a good alternative to these models in modeling survival times of Guinea pigs.

Modeling glass fibers data
This data consists of 63 observations of the strengths of 1.5 cm glass fibres, originally obtained by workers at the UK National Physical Laboratory. This data is given in Appendix (d). These data have also been analyzed by Smith and Naylor (1987). For this data set, we shall compare the fits of the new distribution with some competitive models like OLEW, E-W, T-W and OLL-W. Based on the figures in Table 9 we conclude that the proposed MOLEW model is much better than the OLEW, E-W, T-W, OLL-W models, and a good alternative to these models in modeling glass fibers data.

Concluding remarks
This paper introduces a new four-parameter lifetime model. The new model derives its flexibility and wide applicability from the exponentiated Weibull model. The new density can exhibit many important shapes with different skewness and kurtosis which can be unimodal and bimodal. The new hazard rate can be decreasing, J-hazard rate, bathtub shape (U-hazard rate), constant hazard rate, increasing hazard rate, upside down (reversed bathtub shape) and increasing-constant hazard rate. Various of its structural mathematical properties are derived with details. The new density is expressed as a linear mixture of well-known exponentiated Weibull density. The method of the maximum likelihood is used to estimate the model parameters. Graphical simulation is used in assessing the performance of the estimation method. We proved empirically the importance and flexibility of the new model in modeling various types of data such as failure times, remission times, survival times and strengths data. Many bivariate and the multivariate type versions are derived using the Morgenstern family and Clayton copula. Future works could be allocated to study these new bivariate and the multivariate types.