The Generalized Transmuted Weibull Distribution for Lifetime Data

A new lifetime model, which extends the Weibull distribution using the generalized transmuted-G family proposed by Nofal et al. (2017), called the generalized transmuted Weibull distribution is proposed and studied. Various of its structural properties are derived. The maximum likelihood method is used to estimate the model parameters. A small simulation study is conducted. The new distribution is applied to a real data set to illustrate its flexibility. It can serve as an alternative model to other lifetime models available in the literature for modeling positive real data in many areas.


Introduction
There has been an increased interest among statisticians to develop new methods for generating new families of distributions because there is a persistent need for extending the classical forms of the well-known distributions to be more capable for modeling data in different areas such as lifetime analysis, engineering, economics, finance, demography, actuarial and biological and medical sciences.
In this paper, we define and study a new lifetime model called the generalized transmuted Weibull (GT-W) distribution.Its main feature is that two additional shape parameters are inserted in (1) to provide greater flexibility for the generated distribution.
Based on the generalized transmuted-G (GT-G) family of distributions, we construct the new five-parameter GT-W model and give a comprehensive description of some of its mathematical properties hoping that it will attract wider applications in engineering, survival and lifetime data, reliability and other areas of research.
Henceforth, let G be a continuous baseline distribution.We define the GT-G distribution with two extra parameters  and  by the pdf (3).A random variable  with pdf (3) is denoted by ~GT-G(, , , ).If  =  = 1, it corresponds to the transmuted class (TC) studied by Shaw and Buckley (2007).If  = 1 and  = 0, the GT-G family reduces to the exponentiated-G (E-G) family defined by Gupta et al. (1998) and finally the GT-G family reduces to the baseline distribution when  =  = 1 and  = 0.
Let  be a random variable having the EW distribution with positive parameters ,  and .Then the pdf and cdf of  are given by () =    −1  −()  For further information about the EW distribution we refer to Mudholkar and Srivastava (1993), Mudholkar and Hutson (1996) and Nadarajah and Kotz (2006).
The rest of the paper is outlined as follows.In Section 2, we define the GT-W distribution, provide its sub-models and give some plots for its pdf and hazard rate function (hrf ).We derive useful mixture representations for the pdf and cdf in Section 3.
We provide in Section 4 some mathematical properties of the GT-W distribution including ordinary and incomplete moments, moments of the residual life, reversed residual life, quantile function (qf ), moment generating function (mgf ), Rényi and qentropies and order statistics.The maximum likelihood estimates (MLEs) of the unknown parameters are obtained in Section 5.The simulation results to assess the performance of the proposed maximum likelihood estimation procedure are discussed in Section 6.In Section 7, the GT-W distribution is applied to a real data set to illustrate its potentiality.Finally, in Section 8, we provide some concluding remarks.

The GT-W distribution
By inserting the cdf in (1) in equation ( 2) and omitting the dependence on the parameters , , , , , we obtain the cdf of the five-parameter GT-W model (for  > 0) The corresponding pdf of ( 4) is given by where , ,  and  are positive parameters and || ≤ 1.Here,  is a scale parameter representing the characteristic life, ,  and  are shape parameters representing the different patterns of the GT-W distribution and  is the transmuted parameter.We denote a random variable  having pdf (5) by ~GT-W(, , , , , ).The reliability function (rf), hrf, reversid hazard rate function (rhrf) and cumulative hazard rate function (chrf) of  are given by respectively.
Asymptotics of the cdf, pdf and hrf of the GTW distribution as  → 0 are given by The asymptotics of TGW distribution from cdf, pdf and hrf as  → ∞ are given by The plots of the GT-W density for some parameter values , , ,  and  are displayed in Figure 1. Figure 2 provides some plots of the hrf of the GT-W model for selected parameter values.

Mixture representation
The GT-W density function (5) can be expressed as By inserting (1) in equation ( 6), we obtain So, the GT-W density can be expressed as a mixture of two E-G densities, the first with power parameter  and the seconed with power parameter ( + ).Therefore, equation ( 7) can be expressed as where ℎ  () is the EW density function with power parameter , scale parameter  and shape parameter .then, the GT-W density can be expressed as a mixture of the EW densities and then several of its structural properties can be obtained from (8) and those properties of the EW distribution.
Similarly, the cdf (4) of  can be expressed in the mixture form where   () is the EW cdf with power parameter , scale parameter  and shape parameter .

Mathematical properties
Here, we investigate mathematical properties of the GT-W distribution including ordinary and incomplete moments, moment of the residual life, moment of the reversid residual life, quantile function, mgf and Rényi and q-entropies, order statistics and some characterizations.
Setting  = 1 in (9), we have the mean of .The skewness and kurtosis measures can be calculated from the ordinary moments using well-known relationships.
The th central moment of , say   , follows as The mean, variance, skewness and kurtosis plots of the GT-W are given in Figures 3 and  4, respectively.These plots indicate that the GT-W distribution can model various data types in terms of skewness and kurtosis.
Table 2 provides numerical values for the mean, variance, skewness and kurtosis of  for selected parameter values to illustrate their effects on these measures.

Incomplete moments
The th incomplete moment, say   () of the GT-W distribution is given by   () = ∫  0   ().
The important application of the first incomplete moment is related to the Lorenz and Bonferroni 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.The Lorenz, say   (), and Bonferroni, say [()] curves are respectively defined (see Oluyede and Rajasooriya, 2013) by Another application of the first incomplete moment is related to the mean residual life and the mean waiting time given by  1 () = [1 −  1 ()]/() −  and  1 () =  −  1 ()/(), respectively.
Another interesting function is the mean residual life (MRL) function or the life expectation at age  defined by  1 () = [( − )| > ], which represents the expected additional life length for a unit which is alive at age .The MRL of  can be obtained by setting  = 1 in the last equation.
Therefore, the th moment of the reversed residual life of , given that  > −, becomes where   = (−1)  !/! ( − )!.The mean reversed residual life, also called mean inactivity time (MIT) or mean waiting time (MWT), is given by  1 () = [( − )| ≤ ], and it represents the waiting time elapsed since the failure of an item on condition that this failure had occurred in (0, ).The MIT of  can be obtained simply by setting  = 1 in the above equation.For further information about the properties of the MIT, we refer to Kayid and Ahmad (2004) and Ahmad et al. (2005).

Generating function
Let   () be the mgf of   .Therefore, using (8) the mgf of , say () = (  ), is given by At first, we determine the mgf of (1).We can write this mgf as By expanding exp() and calculating the integral, we have where the gamma function is well-defined for any non-integer .

Rényi and q-entropies
The Rényi entropy of a random variable  represents a measure of variation of the uncertainty.Then, the Rényi entropy of the GT-W distribution is given by By using the pdf in (5), we can write where  =  ( + )/[ (1 + )].
Given that  < 1 and applying a series expansion to , equation ( 12) can be expressed as .
Applying the series expansion to the last equation, we can write where Then, the Rényi entropy of  is given by The q-entropy, say   (), is defined by Hence

Order statistics
Let  1 , … ,   denote  independent and identically distributed GT-W random variables.Further, let  (1) , … ,  () denote the order statistics from these  variables.Then, the pdf of the th order statistic  () , say   (), is given by where and and ℎ  denotes the EW density function with power parameter .Thus, the density function of the GT-W order statistics is a mixture of EW densities.Based on equation ( 13), we can obtain some structural properties of  : from those EW properties.
The th moment of  : (for  > −) is given by Equation ( 14) reveals that The th moment of  : can be expressed as an infinite linear combination of EW moments.

Probability weighted moments
The PWMs are expectations of certain functions of a random variable.They can be derived for any random variable whose ordinary moments exist.The PWM approach can be used for estimating parameters of any distribution whose inverse form cannot be expressed explicitly.

Characterization based on two truncated moments
Here, we provide characterizations of the GT-W distribution in terms of two truncated moments.This characterization result is based on a theorem (see Theorem 1 below) due to Glänzel (1987).The proof of Theorem 1 is given in Glänzel (1990).This result holds also when the interval  is not closed.Moreover, as mentioned above, it could be also applied when the cdf  does not have a closed form.Glänzel (1990) proved that this characterization is stable in the sense of weak convergence.
Theorem 1.Let (Ω, , ) be a given probability space and let  = [, ] be an interval for some  < ( = −∞ ,  = ∞ mightaswellbeallowed).Let : Ω →  be acontinuous random variable with cdf  and let  and ℎ be two real functions defined on  such that is defined with a real function ℎ.Assume that , ℎ ∈  1 (),  ∈  2 () and  is twice continuously differentiable and strictly monotone function on the set . Finally, assume that the equation ℎ =  has no real solution in the interior of .Then  is uniquely determined by the functions , ℎ and , particularly where the function  is a solution of the differential equation  ′ =  ′ ℎ/(ℎ − ) and  is the normalization constant, such that   = 1.

Proposition 1.
Let : Ω → (0, ∞) be a continuous random variable and let The random variable  belongs to GT-W distribution (5) if and only if the function  defined in Theorem 1 has the formand Proof.
Let  be a random variable with density (5), then and finally .
Then, we have Then,  has the pdf (5).
Corollary: Let : Ω → (, ∞) be a continuous random variable and let ℎ() be as in Proposition (1).Then the random variable  has the pdf (5) if and only if the functions  and ℎ defined in Theorem 1 satisfy the following differential equation The general solution of the above differential equation is where  is a constant.There is a set of functions satisfying the differential equation ( 15) is given in Proposition 1 with  = 0.Moreover, there are other triplets (ℎ, , ) satisfying the conditions of Theorem 1.

Estimation
The maximum likelihood method is the most commonly employed method for parameter estimation among several approaches in the literature.The maximum likelihood estimators (MLEs) have desirable properties and can be used when constructing confidence intervals and regions and also in test statistics.The normal approximation for MLEs in large sample distribution theory is easily handled either analytically or numerically.Therefore, we consider the maximum likelihood to estimate the unknown parameters of the GT-W model from complete samples only.Let  1 , … ,   be a random sample of this distribution with unknown parameter vector  = (, , , , ) T .The log-likelihood function for , say ℓ = ℓ(), is given by where   = (  )  ,   = 1 −  −  and   = {(1 + ) − ( + )  }.
Equation ( 16) can be maximized either directly by using the SAS (PROC NLMIXED), R (optim function) or by solving the nonlinear system of equations obtained by differentiating (16).The score vector is given by () = (  ,   ,   ,   ,   ) T .Then, We can obtain the estimates of the unknown parameters by setting the score vector to zero, ( ̂) = 0.By solving these equations simultaneously gives the MLEs  ̂,  ̂,  ̂,  ̂ and  ̂.Statistical software can be used to solve these equations numerically by means of iterative techniques such as the Newton-Raphson algorithm because they can not be solved analytically.For the GT-W distribution all the second order derivatives exist.

Simulation Study
In this section, we conduct a small Monte Carlo simulation based on 3000 Monte Carlo replications.The true parameter values used in the data generating processes are  = 0.1,  = 0.5,  = 1,  = 7.3 and  = −0.8.Different sample sizes  = 50, 60, 70, 80, 90, 100, 150, 200 and 500 were considered.The mean estimate, bias and the root-meansquare error (RMSE) of the parameter estimates for the maximum likelihood estimates were determined from this simulation study and are presented in Table 2.It can be seen that the estimates are stable and quite close the true parameter values for these sample sizes.Furthermore, as the sample size increases the RMSE decreases in all cases.

Application
In this section, we provide an application of the GT-W distribution to show the importance of the new model.We now provide a data analysis in order to assess the goodness-of-fit of the proposed model.We will make the use of the data set on the remission times (in months) of a random sample of 128 bladder cancer patients (Lee and Wang, 2003) is given by: 0.   2 ) }.
• ETGR: The parameters of the above densities are all positive real numbers except for the TLE, TMW, TAW and ETGR distributions for which || ≤ 1 and 0<  <  (or 0<  < ) for the TAW.
In order to compare the fitted models, we consider some goodness-of-fit measures including the Akaike information criterion (), consistent Akaike information criterion () and −2ℓ ̂, where ℓ ̂ is the maximized log-likelihood,  = −2ℓ ̂+ 2,  = −2ℓ ̂+ 2/( −  − 1),  is the number of parameters and  is the sample size.Moreover, we use the Anderson-Darling ( * ) and the Cramér-von Mises ( * ) statistics in order to compare the fits of the two new models with other nested and non-nested models.The 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 respectively,   = (  ), where the   's values are the ordered observations.The smaller these statistics are, the better the fit.Upper tail percentiles of the asymptotic distributions of these goodness-of-fit statistics were tabulated in Nichols and Padgett (2006).
Table 4 lists the values of −2ℓ ̂, , ,  * and  * whereas the MLEs, their corresponding standard errors, of the model parameters are given in Table 5.These numerical results are obtained using the MATH-CAD PROGRAM.
The fitted pdf, estimated cdf and QQ-plot of the GT-W model are displayed in Figures 5  and 6, respectively.
In Table 4, we compare the fits of the GT-W model with the Mc-W, TLE, TMW, MBW, TAW, ETGR and W models.We note that the GT-W model has the lowest values for the −2ℓ ̂, , ,  * and  * statistics among all fitted models.So, the GT-W model could be chosen as the best model.It is quite clear from the figures in Table 2 and Figures 3 and 4, that the GT-W distribution can provide the best fits to these data.So, we prove that this new model can be better model than other competitive lifetime models.We provide some of its mathematical and statistical properties.The GT-W density function can be expressed as a mixture of EW densities.We derive explicit expressions for the ordinary and incomplete moments, quantile and generating functions, probability weighted moments, Rényi and q-entropies and order statistics.We discuss maximum likelihood estimation.The proposed distribution applied to a real data set provides better fits than some other nested non-nested models.We hope that the proposed model will attract wider application in areas such as engineering, reliability, survival and lifetime data, hydrology, economics and others.

Appendix
The elements of the observed information matrix are given by

Figure 1 :
Figure 1: Plots of the GT-W density function for some parameter values.

Figure 2 :
Figure 2: Plots of the GT-W hazard rate function for some parameter values

Figure 3 :Figure 4 :
Figure 3: Plots of mean and variance of the GT-W distribution for several values of parameters

Figure 5 :Figure 6 :
Figure 5: The fitted pdf and estimated cdf of the GT-W model

Table 3 : Mean estimates, bias and root mean squared errors of
, , ,  and

Table 3 : Mean estimates, bias and root mean squared errors of
, , ,  and