The Transmuted Exponentiated Generalized-G Family of Distributions

We introduce a new class of continuous distributions called the transmuted exponentiated generalized-G family which extends the exponentiated generalized-G class introduced by Cordeiro et al. (2013). We provide some special models for the new family. Some of its mathematical properties including explicit expressions for the ordinary and incomplete moments, generating function, Rényi and Shannon entropies, order statistics and probability weighted moments are derived. The estimation of the model parameters is performed by maximum likelihood. The flexibility of the proposed family is illustrated by means of an application to a real dataset.


Introduction
In the last few decades, there have been an increased interest among statisticians in defining new generators of univariate distributions, by adding one or more shape parameter(s) to a baseline distribution, to provide great flexibility in modelling data in several applied areas such as reliability, engineering, economics, biological studies, environmental sciences, finance and medical sciences. Some well-known generators are the Marshall-Olkin-G (MO-G) by Marshall and Olkin (1997), the beta-G (B-G) by Eugene et al. (2002), the Kumaraswamy-G (Kw-G) by Cordeiro and de Castro (2011), the McDonald-G (Mc-G) by Alexander et al. (2012), the gamma-G by Zografos and Balakrishanan (2009), the Kumaraswamy odd log-logistic-G (KwOLL-G) by , the beta odd log-logistic generalized by Cordeiro et al. (2015), the Kumaraswamy transmuted-G (KwT-G) by , the beta transmuted-G (BT-G) by  and the generalized transmuted-G (GT-G) by Nofal et al. (2015).
Further, Shaw and Buckley (2007) introduced an interesting technique of adding a new parameter to an existing distribution called the transmuted-G (TG for short) family. Let ( ) be a baseline cumulative distribution function (cdf) and ( ) be its probability density function (pdf) with a parameter vector . Shaw and Buckley (2007)  This vast amount of literature motivated us to use the TG family to construct a new generator called the transmuted exponentiated generalized-G (TExG-G for short) family. We provide a comprehensive description of their properties with the hope that the TExG-G class will attract wider applications in biology, medicine, economics, reliability, engineering, and in other areas of research. In this article, we define the TExG-G family using the pdf and cdf of the exponentiated generalized-G (ExG-G) family proposed by Cordeiro et al. (2013). The cdf and pdf of the ExG-G class are given, respectively, by and where and are positive shape parameters.
The rest of the paper is outlined as follows. In Section 2, we define the TExG-G family of distributions and provide its special models. In Section 3, we derive a very useful linear representation for the TExG-G density function. Three special models of this family are presented in Section 4 and some plots of their pdf's are given. We obtain in Section 5 some general mathematical properties of the proposed family including asymptotics, extreme values, ordinary and incomplete moments, probability weighted moments (PWMs), mean deviations, residual life function and reversed residual life function. Order statistics and their moments are investigated in Section 6. In Section 7, we determine the stress-strength model for the proposed family. In Section 8, some characterizations results are provided. Maximum likelihood estimation (MLE) of the model parameters is investigated in Section 9. In Section 10, we perform an application to a real dataset to illustrate the potentiality of the new family. Section 11 deals with a small simulation study to assess the performance of the MLE method. Finally, some concluding remarks are presented in Section 12.

The TExG-G Family
The cdf of the TExG-G family using (3), is defined by The pdf corresponding to (5) is defined by We denote by TExG-G( ) a random variable X with the pdf (6 Some special classes of the TExG-G family are listed in Table 1.

Linear Representation
In this section, we provide a useful representation for the cdf and the pdf of TExG-G family.
Using the series expansion The cdf in (5) can be expressed as The cdf of TExG-G family in (7) can be expressed as where ( ) ( ) is the cdf of the Ex-G family with power parameter and The corresponding TExG-G density function is obtained by differentiating (8) where ( ) ( ) ( ) is the Ex-G pdf with power parameter .

Special Models
In this section, we provide some examples of the TExG-G family of distributions. The pdf (6) will be most tractable when ( ) and ( ) have simple analytic expressions. These special models generalize some well-known distributions in the literature. Now, we provide three special models of this family corresponding to the baseline Pareto (Pa), Weibull (W) and Fréchet (Fr) distributions.

The TExG-Pareto (TExGPa) Distribution
The Pareto (Pa) distribution with shape parameter and scale parameter has pdf and cdf given by ( ) (for ) and ( ) ( ) respectively. Then, the pdf of the TExGPa model is given by The TExGPa distribution becomes the TPa distribution when . For , the TExGPa distribution reduces to the ExGPa model. For and we obtain the TExPa and TGPa distributions, respectively. The plots of the density function of the TExGPa distribution are displayed in Figure 1 for selected parameter values.

The TExG-Weibull (TExGW) Distribution
The Weibull distribution with scale parameter and shape parameter has pdf and cdf given by ( ) ( ) (for ) and ( ) ( ) , respectively. Then, the pdf of the TExGW model is given by The TExGW distribution becomes the TW distribution when . For , the TExGW distribution reduces to the ExGW model. For and we obtain the TExW and TGW distributions, respectively. The plots of the density function of the TExGW distribution are displayed in Figure 2 for selected parameter values.

The TExG-Fréchet (TExGFr) Distribution
The Fréchet distribution with positive parameters and has pdf and cdf given by The TExGFr distribution becomes the TFr distribution when . For , the TExGFr distribution reduces to the ExGFr model. For and we obtain the TExFr and TGFr distributions, respectively. The plots of the density function of the TExGFr distribution are displayed in Figure 2 for selected parameter values.

Mathematical Properties
In this section, we investigate mathematical properties of the TExG-G family of distributions. Established algebraic expansions to determine some structural properties of the TExG-G family of distributions can be more efficient than computing those directly by numerical integration of its density function.

Asymptotics
The asymptotics of ( ), ( ) and ( ) as ( ) are given by where ( ) is the hazard rate function (hrf) of the TExG-G family.

Extreme Values
If ( ) denotes the mean of a random sample from (6), then by the usual central limit theorem √ ( ( )) √ ( ) approaches the standard normal distribution as under suitable conditions. Sometimes one would be interested in the asymptotes of the extreme values ( ) and ( ).
First, suppose that G belongs to the max domain of attraction of the Gumbel extreme value distribution. Then by Leadbetter et al. (1987), there must exist a strictly positive function, say ( ), such that

Moments
The th ordinary moment of is given by Using (5), we obtain Hereafter, denotes the Ex-G distribution with power parameter( ). Setting in (10), we have the mean of .
The last integration can be computed numerically for most parent distributions. The skewness and kurtosis measures can be calculated from the ordinary moments using wellknown relationships.
The th central moment of , say , follows as where , etc. The skewness and kurtosis measures also can be calculated from the ordinary moments using well-known relationships.
The moment generating function (mgf) of , say ( ) ( ) is given by

Incomplete Moments
The main application of the first incomplete moment refers to the Bonferroni and Lorenz curves. These curves are very useful in economics, reliability, demography, insurance and medicine. The answers to many important questions in economics require more than just knowing the mean of the distribution, but its shape as well. This is obvious not only in the study of econometrics but in other areas as well. The th incomplete moments, say ( ) is given by Using equation (9), we obtain The first incomplete moment of the TExG-G family, ( ), can be obtained by setting in (11).
Another application of the first incomplete moment is related to meanresidual life and mean waiting time given by ( ) , ( )-( ) and ( ) , ( ) ( )respectively.

Probability Weighted Moments
The PWMs are expectations of certain functions of a random variable and they can be defined for any random variable whose ordinary moments exist. The PWM method can generally be used for estimating parameters of a distribution whose inverse form cannot be expressed explicitly.
The ( )th PWM of following the TExG-G family, say , is formally defined by Using equations (5) and (6) After some algebra, we can write Then, the ( )th PWM of can be expressed as ∑ ( )

Mean Deviations
The mean deviations about the mean , (| |)and about the median , (| |)of are given by ( ) ( ) and ( ), respectively, where ( ), ( ) ( ) is the median, ( ) is easily calculated from (5) and ( ) is the first incomplete moment given by (11) with . Now, we provide two ways to determine and . First, a general equation for ( ) can be derived from (11) as where ( ) ∫ ( ) is the first incomplete moment of the exp-G distribution. A second general formula for ( ) is given by can be computed numerically.
These equations for ( ) can be used to construct Bonferroni and Lorenz curves defined for a given probability by ( ) ( ) ( ) and ( ) ( ) , respectively, where ( ) and ( ) is the qf of at .

Residual Life and Reversed Residual Life Functions
The th moment of the residual life, say ( ) ,( ) | -, ,..., uniquely determine ( ). The th moment of the residual life of is given by Another interesting function is the mean residual life (MRL) function or the life expectation at age defined by ( ) ,( )| -, which represents the expected additional life length for a unit which is alive at age . The MRL of can be obtained by setting in the last equation.

Order Statistics
Order statistics make their appearance in many areas of statistical theory and practice. Let be a random sample from the TExG-G family of distributions and let ( ) ( ) be the corresponding order statistics. The pdf of th order statistic, say , can be written as where ( ) is the beta function.
Substituting (5) and (6) in equation (12) and using a power series expansion, we get Moreover, the pdf of can be expressed as Then, the density function of the TEx-G order statistics is a mixture of Ex-G densities. Based on the last equation, we note that the properties of follow from those properties of . For example, the moments of can be expressed as Based upon the moments in equation (13), we can derive explicit expressions for the Lmoments of as infinite weighted linear combinations of the means of suitable TExG-G order statistics. They are linear functions of expected order statistics defined by ∑ ( ) . / ( )

Stress-Strength Model
Stress-strength model is the most widely approach used for reliability estimation. This model is used in many applications of physics and engineering such as strength failure and system collapse. The reliability, say , where ( ) is a measure of reliability of the system when it is subjected to random stress and has strength .
The system fails if and only if the applied stress is greater than its strength and the component will function satisfactorily whenever .
can be considered as a measure of system performance and naturally arises in electrical and electronic systems. Other interpretation can be that, the reliability of the system is the probability that the system is strong enough to overcome the stress imposed on it.
Let and be two independent random variables have TExG-G( ) and TExG-G( ) distributions. From (10) and (9), the pdf of and the cdf of can be, respectively, expressed by Then is given by

Characterizations
In this section, we present certain characterizations of TExG-G distribution. The first characterization is based on a simple relationship between two truncated moments. It should be mentioned that for this characterization, the cdf need not have a closed form. We believe, due to the nature of the cdf of TExG-G class, there may not be other possibly interesting characterizations than the ones presented in this section. Our first characterization result borrows from a theorem due to (Glänzel,1987), see Theorem 8.1 below. Note that the result holds also when the interval is not closed. Moreover, as shown in (Glänzel,1990), this characterization is stable in the sense of weak convergence.
Theorem 8.1. Let ( ) be a given probability space and let , be an interval for some ( might as well be allowed). Let be a continuous random variable with the distribution function and let and be two real functions defined on such that is defined with some real function . Assume that ( ), ( ) and is twice continuously differentiable and strictly monotone function on the set . Finally, assume that the equation has no real solution in the interior of . Then is uniquely determined by the functions , and , particularly where the function is a solution of the differential equation and is the normalization constant, such that ∫ .
Here is our first characterization of TExG-G distribution.
Conversely, if is given as above, then and hence The general solution of the differential equation in Corollary 8.1 is where is a constant. Note that a set of functions satisfying the differential equation in Corollary 8.1, is given in Proposition 8.1 with However, it should be also noted that there are other triplets ( ) satisfying the conditions of Theorem 8.1.

Estimation
Let be a random sample from the TExG-G distribution with parameters and . Let ( ) be the parameter vector. For determining the MLE of , we have the log-likelihood function The components of the score vector, ( ) ( ) are given by Setting the nonlinear system of equations and and solving them simultaneously yields the MLE ̂ ( ̂ ̂ ̂ ̂ ) . To solve these equations, it is usually more convenient to use nonlinear optimization methods such as the quasi-Newton algorithm to numerically maximize . For interval estimation of the parameters, we obtain the observed information matrix ( ) * + (for ), whose elements can be computed numerically.
Under standard regularity conditions when , the distribution of ̂ can be approximated by a multivariate normal ( ( ̂) ) distribution to construct approximate confidence intervals for the parameters. Here, ( ̂) is the total observed information matrix evaluated at ̂. The method of the re-sampling bootstrap can be used for correcting the biases of the MLEs of the model parameters. Good interval estimates may also be obtained using the bootstrap percentile method. The elements of ( ) are given in Appendix A.

Application
In this section, we demonostrate empirically the potentiality of the TExGW distribution presented in Section 4 by means of an application to a real data. The MLEs of the model parameters and some goodness-of-fit statistics for the fitted models are computed using MATH-CAD.
The data set refers to the remission times (in months) of a random sample of 128 bladder cancer patients (Lee and Wang, 2003). These data have been used by Nofal et al. (2015) and Mead and Afify (2015) to fit the generalized transmuted log logistic and Kumaraswamy exponentiated Burr XII distributions, respectively. We compare the fit of the TExGW distribution with those of the transmuted Weibull Lomax (TWL) ( Table 2 lists the numerical values of  ̂,  ,  ,  ,  ,  and for the models fitted. The MLEs and their corresponding standard errors (in parentheses) of the model parameters are given in Table 3.
In Table 2

Simulation Study
In Table 4, we conducted simulation study to assess the performance of the maximum likelihood estimation procedure for estimating the TExGW distribution parameters using Monte Carlo simulation. Samples of sizes 100, 200, 500 and 1000 are generated for diferent combinations of parameters ( , , , and ) from TExGW distribution. We repeated the simulation k =100 times and calculated the MLEs and the standard deviations of the parameter estimates. The empirical results are given in Table 4 shows that the estimates are quite stable and are close to the true value of the parameters for these sample sizes.

Conclusions
In many applied areas there is a clear need for extended forms of the well-known distributions. Generally, the new distributions are more flexible to model real data that present a high degree of skewness and kurtosis. We propose a new transmuted exponentiated generalized-G (TExG-G) family of distributions, which extends the exponentiated generalized-G (ExG-G) family (Cordeiro et al., 2013) by adding one extra shape parameter. Many well-known models emerge as special cases of the ExG-G family by using special parameter values. Some mathematical properties of the new class including explicit expansions for the ordinary and incomplete moments, quantile and generating functions, mean deviations, entropies, order statistics and probability weighted moments are provided. The model parameters are estimated by the maximum likelihood estimation method and the observed information matrix is determined. We perform a Monte Carlo simulation study to assess the finite sample behavior of the maximum likelihood estimators. We prove empirically by means of an application to a real data set that special cases of the proposed family can give better fits than other models generated from well-known families.

Appendix A
The elements of the observed matrix ( ) are given below: