The Burr X Exponentiated Weibull Model: Characterizations, Mathematical Properties and Applications to Failure and Survival Times Data

In this article, we introduce a new three-parameter lifetime model called the Burr X exponentiated Weibull model. The major justification for the practicality of the new lifetime model is based on the wider use of the exponentiated Weibull and Weibull models. We are motivated to propose this new lifetime model because it exhibits increasing, decreasing, bathtub, J shaped and constant hazard rates. The new lifetime model can be viewed as a mixture of the exponentiated Weibull distribution. It can also be viewed as a suitable model for fitting the right skewed, symmetric, left skewed and unimodal data. We provide a comprehensive account of some of its statistical properties. Some useful characterization results are presented. The maximum likelihood method is used to estimate the model parameters. We prove empirically the importance and flexibility of the new model in modeling two types of lifetime data. The proposed model is a better fit than the Poisson Topp Leone-Weibull, the Marshall Olkin extended-Weibull, gamma-Weibull , Kumaraswamy-Weibull , Weibull-Fréchet, beta-Weibull, transmuted modified-Weibull, Kumaraswamy transmutedWeibull, modified beta-Weibull, Mcdonald-Weibull and transmuted exponentiated generalized-Weibull models so it is a good alternative to these models in modeling aircraft windshield data as well as the new lifetime model is much better than the Weibull-Weibull, odd WeibullWeibull, Weibull Log-Weibull, the gamma exponentiated-exponential and exponential exponential-geometric models so it is a good alternative to these models in modeling the survival times of Guinea pigs. We hope that the new distribution will attract wider applications in reliability, engineering and other areas of research.


Introduction
It is known that the Weibull distribution has been the most popular distribution for modeling lifetimes (see Murthy et al., 2004 andRinne, 2009) and has been extensively used for modeling data in engineering, reliability and biological researches. The major weakness of this distribution is its inability to accommodating nonmonotone hazard rates. This has led to the need of exploring more generalizing of this model. The first generalization allowing for nonmonotone hazard rates is the exponentiated Weibull (EW) model (see Mudholkar and Srivastava (1993) and Mudholkar et al. (1995)). The goal of this paper is to introduce a new extremely flexible version of the EW model.
To this end we will use the BrX-G for generating the new extreme flexible version of the EW model.
This paper is organized as follows. In Section 2, we define the new distribution. Section 3 deals with some characterizations of the new model. we derive some of its mathematical properties in Section 4. The maximum likelihood method is presented in Section 5. In Section 6, we illustrate the importance of the new model by means of two applications to real data sets. The paper is concluded in Section 7.

The new model and its justification
By inserting ( ; , ) in (1) we obtain the cdf of the Burr X exponentiated Weibull (BrXEW) model as The corresponding pdf is ( ; , , ) Now, we provide a very useful linear representation for the BrXEW density function. If | | < 1 and > 0 is a real non-integer, the following power series holds (5) Applying (5) to the term A in (4) we have Applying the power series to the term B in (6) we have Consider the series expansion , | | < 1, > 0.
(8) Applying the expansion (8) to (7) to C in (7) we arrive at and (2 + +2) ( ) is the cdf of the EW model with power parameter (2 + + 2) . Equation (5) reveals that the density of can be expressed as a linear mixture of EW densities. So, several mathematical properties of the new family can be obtained from those of the exp-Li distribution. Similarly, the cdf of the BrXEW model can also be expressed as a mixture of EW cdfs given by where (2 + +2) ( ) is the cdf of the EW model with power parameter (2 + + 2) .
The justification for the practicality of the BrXEW lifetime model is based on the wider use of the EW and W models. We are also motivated to introduce the BrXEW lifetime model since it exhibits increasing, decreasing, bathtub, J shaped, and constant hazard rates as illustrated in Figure 2 (b1 to b5, respectively). We mentioned before that the BrXEW lifetime model can be viewed as a mixture of the EW distribution. It can be considered as a suitable model for fitting the right skewed, symmetric, left skewed and unimodal data. The proposed BrXEW lifetime model is a much better fit than the Poisson Topp Leone-Weibull, the Marshall Olkin extended-Weibull, gamma-Weibull, Kumaraswamy-Weibull, Weibull-Fréchet, beta-Weibull, transmuted modified-Weibull, Kumaraswamy transmuted-Weibull, modified beta-Weibull, Mcdonald-Weibull and transmuted exponentiated generalized-Weibull models, so the new lifetime model is a good alternative to these models in modeling aircraft windshield data. It is also a much better fit than the Weibull-Weibull, odd Weibull-Weibull, Weibull Log-Weibull, the gamma exponentiated-exponential and exponential exponential-geometric models, so it is a good alternative to these models in modeling the survival times of Guinea pigs.

Characterizations results
This section is devoted to the characterizations of the BrXEW distribution in different directions: ( ) based on the ratio of two truncated moments; ( ) in terms of the hazard function; ( ) in terms of the reverse hazard function and ( ) based on the conditional expectation of certain function of the random variable. Note that ( ) can be employed also when the cdf does not have a closed form. We would also like to mention that due to the nature of BrXEW distribution, our characterizations may be the only possible ones. We present our characterizations ( ) − ( ) in four subsections.

Characterizations based on two truncated moments
This subsection deals with the characterizations of BrXEW distribution based on the ratio of two truncated moments. Our first characterization employs a theorem due to Glänzel (1987), see Theorem 1 (see Hamedani et al. (2018) and Hamedani et al. (2019)). The result, however, holds also when the interval is not closed, since the condition of the Theorem is on the interior of .
Proposition 3.1. Let : → (0, ∞) be a continuous rv and let The rv has pdf (4) if and only if the function defined in Theorem 1 is of the form Proof. Suppose the rv has pdf (4) , then Further, Conversely, if is of the above form, then , > 0, > 0. Now, according to Theorem 1, has density (4).
where is a constant. We like to point out that one set of functions satisfying the above differential equation is given in Proposition 3.1 with = 1 2 .

Characterization in terms of hazard function
The hazard function, ℎ , of a twice differentiable distribution function, , satisfies the following first order differential equation It should be mentioned that for many univariate continuous distributions, the above equation is the only differential equation available in terms of the hazard function. In this subsection we present non-trivial characterizations of BrXEW distribution, for = 1 , in terms of the hazard function.
Proposition A.2. Let : → (0, ∞) be a continuous random variable. The rv has pdf (4) if and only if its hazard function ℎ ( ) satisfies the following differential equation Proof. If has pdf (4) , then clearly the above differential equation holds. If the differential equation holds, then from which we arrive at the hazard function of (4) when = 1.

Characterization in terms of the reverse hazard function
The reverse hazard function, , of a twice differentiable distribution function, , is defined as In this subsection we present a characterization of BrXEW distribution in terms of the reverse hazard function.

Proposition 3.3.
Let : → (0, ∞) be a continuous random variable. The rv has pdf (4) if and only if its reverse hazard function ( ) satisfies the following differential equation Proof. Is similar to that of Proposition 3.2.

Characterization based on the conditional expectation of certain function of the random variable
In this subsection we employ a single function (or 1 ) of and characterize the distribution of in terms of the truncated moment of ( ) (or 1 ( )). The following propositions have already appeared in Hamedani's previous work (2013), so we will just state them here which can be used to characterize BrXEW distribution. and 1 = 1+ . Proposition 3.5 provides a characterization of BrXEW distribution.

Mathematical properties 4.1 Moments
The th ordinary moment of is given by Setting = 1 in (11), we have the mean of .

Generating function
Using the series expansion
The first incomplete moment is obtained by setting = 1 in ( ) .

Residual and reversed residual life
The

Stress-strength model
In stress-strength modeling, 1 , 2 = ( 2 < 1 ) is a measure of reliability of the system when it is subjected to random stress 2 and has strength 1 . The system fails if and only if the applied stress is greater than its strength and the components will function satisfactorily whenever 1 > 2 .
1 , 2 can be considered as a measure of system performance and naturally arise in electrical and electronic systems. Other interpretation can be given as the reliability 1 , 2 of a system is the probability that the system is strong enough to overcome the stress imposed on it. Let 1 and 2 be two independent random variables with BrXEW ( 1 , , ) and BrXEW ( 2 , , ) distributions, respectively. The reliability is defined by 1 , 2 | 2 < 1 = ∫ 1 ( ; 1 , , )

Order statistics
Order statistics make their appearance in many areas of statistical theory and practice. Let 1 : , … , : be a random sample from the BrXEW distribution and let (1) , … , ( ) be the corresponding order statistics. The pdf of th order statistic, say : , can be written as

Parameter Estimation
Several methods for parameter estimation were proposed in the literature but the maximum likelihood method is the most commonly employed. So, we consider the estimation of the unknown parameters of this family from complete samples only by maximum likelihood method. Let 1 , … , be a random sample from the BrXEW distribution with parameters , and . Let = ( , , ) be the 3 × 1 parameter vector. For determining the MLE of , we have the log-likelihood function

Applications
In this section, we provide two applications of the BrXEW distribution to show empirically its potentiality. In order to compare the fits of the BrXEW distribution with other competing distributions, we consider the Cramér-von Mises ( * ) and the Anderson-Darling ( * ) statistics. These two statistics are widely used to determine how closely a specific cdf fits the empirical distribution of a given data set.  Table 1 reveal that the BrXEW distribution yields the lowest values of these statistics and hence provides the best fit to the two data sets.

Survival times (in days) of 72 Guinea pigs
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 EE-Gc: and a good alternative to these models. In Application 2, the proposed BrXEW lifetime model is much better than the W-W, OW-W, WLog-W, GaE-E, EE-Gc models, and a good alternative to these models.

Conclusions
In this article, we introduce a new three-parameter lifetime model called the Burr X exponentiated Weibull model. The major justification for the practicality of the new lifetime model is based on the wider use of the exponentiated Weibull and Weibull models. We are motivated to propose this new lifetime model because it exhibits increasing, decreasing, bathtub, J shaped and constant hazard rates. The new lifetime model can be viewed as a mixture of the exponentiated Weibull distribution. It can also be viewed as a suitable model for fitting the right skewed, symmetric, left skewed and unimodal data. We provide a comprehensive account of some of its statistical properties also some useful characterization results are presented. The maximum likelihood method is used to estimate the model parameters. We prove empirically the importance and flexibility of the new model in modeling two types of lifetime data. The proposed BrXEW lifetime model is a much better fit than the Poisson Topp Leone-Weibull, the Marshall Olkin extended-Weibull, gamma-Weibull, Kumaraswamy-Weibull, Weibull-Fréchet, beta-Weibull, transmuted modified-Weibull, Kumaraswamy transmuted-Weibull, modified beta-Weibull, Mcdonald-Weibull and transmuted exponentiated generalized-Weibull models, so the new lifetime model is a good alternative to these models in modeling aircraft windshield data. It is also a much better fit than the Weibull-Weibull, odd Weibull-Weibull, Weibull Log-Weibull, the gamma exponentiated-exponential and exponential exponential-geometric models, so it is a good alternative to these models in modeling the survival times of Guinea pigs. We hope that the new model will attract wider applications in reliability, engineering and other areas of research.