A New Extremely Flexible Version of The Exponentiated Weibull Model: Theorem and Applications to Reliability and Medical Data Sets

Abstract In this work, a new lifetime model is introduced and studied. The major justification for the practicality of the new model is based on the wider use of the exponentiated Weibull and Weibull models. We are also motivated to introduce the new lifetime model since it exhibits decreasing, upside down-increasing, constant, increasing-constant and J shaped hazard rates also the density of the new distribution exhibits various important shapes. The new model can be viewed as a mixture of the exponentiated Weibull distribution. It can also be considered as a suitable model for fitting the symmetric, left skewed, right skewed and unimodal data. The importance and flexibility of the new model is illustrated by four read data applications.


Justification
The major justification for the practicality of the new model is based on the wider use of the exponentiated Weibull and Weibull models.We are also motivated to introduce the new lifetime model since it exhibits decreasing, unimodal and constant hazard rates (see Figure 2) also the P.D.F. of the new distribution exhibits various important shapes such as decreasing, unimodal, right skewed and left skewed (see Figure 1).  1 shows that the P.D.F.BHEW distribution exhibits various important shapes such as decreasing, unimodal, right skewed and left skewed, from Figure 2 we conclude that the H.R.F. of the BHEW distribution exhibits decreasing, unimodal and constant hazard rates (Figure 1 and 2 are given in Appendix).

𝑑
. Then, Eq. ( 2) can be writen as where  0 =    0 and, for  ≥ 1, we have At the end, the C.D.F. ( 2) can be written as where  0 = 1 −   , for  ≥ 1 we have  0 = −  and is the C.D.F. of the Exp-G family with power parameter ( + 1) .By differentiating (6), we obtain the same mixture representation where is the EW P.D.F. with power parameter ( + 1) .Eq. ( 7) means that the BHEW function is a linear combination of EW densities.So that, some the structural properties of the new model can be immediately obtained from the well-established properties of the EW distribution.

Effect of on the mean, variance, skewness and kurtosis
From Table 1 we note that: 1- decreases as  increases.
2-The variance decreases as  increases.
The  ℎ incomplete moment of  is defined by We can write from ( 7) By setting  = 1,2,3 and 4 we get and

Estimation
Let  1 , … ,   be a R.S. from the BHEW distribution with parameters ,  and  .Let  be the 3 × 1 parameter vector.For getting the maximum likelihood estimates (M.L.E.) of  , we have the log-likelihood (L.L.) function The components of the score vector is available if needed.

Applications
In this Section, we provide four applications to show empirically its potentiality.We consider the Cramér-Von Mises W * and the Anderson-Darling  * statistics.The computations are carried out using the R software.The M.L.E. and the corresponding standard errors (S.E.) (in parentheses) of the new model parameters are given in Tables 2,  4, 6, and 8.The numerical values of the W * and A * are listed in Tables 3, 5, 7, and 9.The estimated P.D.F., P-P plot, TTT plot and Kaplan-Meier survival plot of the four data sets of the proposed model are displayed in Figures 3, 4,5 and 6.These four data sets were used for fitting the Odd Lindley EW by Aboraya (2018).

Application 1
The data consist of 84 observations.The data are: 0.040,  The parameters of the above densities are all positive real numbers except for the TM-W and TExG-W distributions.Tables 2 list the values of above statistics for seven fitted models.The M.L.E.s and their corresponding standard errors (in parentheses) of the model parameters are also given in these tables.The figures in Table 3 reveal that the new distribution yields the lowest values of these statistics and hence provides the best fit to the two data sets.

Application 2
This data set represents the remission times (in months) of a random sample of 128 bladder cancer patients.This data is given by: 0.

Application 3
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) We shall compare the fits of the BHEW distribution with those of other competitive models, namely: the Weibull-Weibull (W-W), the Odd Weibull-Weibull (OW-W), the gamma exponentiated-exponential (GaE-E) distributions, whose P.D.F.s (for x > 0 ) (for more details about these P.D.F.s see Aboraya ( 2018)).

Figure 3 :
Figure 3: Estimated P.D.F., P-P plot, TTT plot and Kaplan-Meier survival plot for data set I.

Figure 4 :
Figure 4: Estimated P.D.F., P-P plot, TTT plot and Kaplan-Meier survival plot for data set II.

Figure 6 :
Figure 6: Estimated P.D.F., P-P plot, TTT plot and Kaplan-Meier survival plot for data set IV.
The new model can be viewed as a mixture of the EW distribution.It can also be considered as a suitable model for fitting the symmetric, left skewed, right skewed, and unimodal data (see application section).

Table 3 :
W  and A  for data set I.

Table 5 :
W  and A  for data set II.