The gamma extended Weibull distribution
 Gauss M. Cordeiro^{1},
 Maria do Carmo S. Lima^{1},
 Antonio E. Gomes^{2}Email author,
 Cibele Q. daSilva^{2} and
 Edwin M. M. Ortega^{3}
https://doi.org/10.1186/s4048801600432
© Cordeiro et al. 2016
Received: 10 September 2015
Accepted: 9 February 2016
Published: 15 March 2016
Abstract
The twoparameter Weibull has been the most popular distribution for modeling lifetime data. We propose a fourparameter gamma extended Weibull model, which generalizes the Weibull and extended Weibull distributions, among several other models. We obtain explicit expressions for the ordinary and incomplete moments, generating and quantile functions and mean deviations. We employ the method of maximum likelihood for estimating the model parameters. We propose a loggamma extended Weibull regression model with censored data. The applicability of the new models is well justified by means of two real data sets.
Keywords
Extended Weibull distribution Gamma extended Weibull distribution Generating function Maximum likelihood estimation Mean deviation Quantile functionMSC
62E10 62E20 62F10Introduction
There are hundreds of continuous univariate distributions and recent developments focus on constructing general distributions from classic ones. Many distributions have been used as models to make inferences about a population based on a set of empirical data from that population. Determining an adequate model to make inferences is a very important problem. The Weibull distribution is commonly used for modeling systems with monotone failure rates. However, the data sets in reliability analysis, especially over the lifecycle of the product, can involve high initial failure rates and eventual high failure rates due to aging and wear out, indicating a bathtub failure rate. The distributions that allow only monotone failure rates might not be adequate for modeling populations giving rise to such data. The major weakness of the Weibull distribution is its inability to accommodate nonmonotone hazard rates, which has led to new generalizations of this distribution. One of the first extensions allowing for nonmonotone hazard rates, including the bathtub shaped hazard rate function (hrf), is the exponentiated Weibull (ExpW) (Mudholkar and Srivastava 1993; Mudholkar et al. 1995; and Mudholkar et al. 1996) distribution. It has been well established in the literature that the ExpW distribution provides significantly better fits than the wellknown exponential, gamma, Weibull and lognormal distributions. In the last paper, the authors presented a more flexible threeparameter extended Weibull (EW) model. Further, Shao et al. (2004) used this distribution to study flood frequency and Hao and Singh (2008) described some of its applications in hydrology. In this paper, we propose a further generalization by taking the EW distribution as the baseline model.
respectively, where λ>0 is a scale parameter and α≥0 and β>0 are shape parameters. The support of the EW distribution is \(\mathbb {R}^{+}\). The forms of the pdf and cdf when α goes to zero tend to those ones when α=0. Clearly, the cdf (2) extends the Weibull cdf and this fact justifies the terminology EW model. Due to the shape parameter α, more flexibility can be incorporated in model (1), which is very useful for lifetime data. The survival function associated to (1) is S _{ λ,α,β }(x)=1−G _{ λ,α,β }(x) for α>0 and S _{ λ,β }(x)=1−G _{ λ,β }(x) for α=0.
respectively, where g(x)=d G(x)/d x, \(\Gamma (a) = \int _{0}^{\infty }t^{a1}\,{\mathrm {e}}^{t}dt\) is the gamma function, \(\gamma (a,z) = {\int _{0}^{z}} t^{a1}\,{\mathrm {e}}^{t}dt\) denotes the incomplete gamma function and γ _{1}(a,z)=γ(a,z)/Γ(a) is the incomplete gamma function ratio.
The GG model has the same parameters of the parent G distribution plus one extra shape parameter a>0. Each new GG distribution can be determined from a specified G model. For a=1, the G distribution is a basic exemplar with a continuous crossover towards cases with different shapes (for example, a particular combination of skewness and kurtosis).
We introduce a new fourparameter model called the “gamma extended Weibull” (“GEW”) distribution, which contains as special models some distributions such as the EW distribution. In fact, this model represents only a basic exemplar of the GEW distribution. We also study some of its mathematical properties. The paper is outlined as follows. In Section 2, we define the GEW distribution and provide some of its special cases. Further, two useful expansions for its density and cumulative distributions are derived in Section 3. In Section 4, we obtain its quantile function (qf). The ordinary moments and mean deviations are investigated in Section 5. Maximum likelihood estimation of the model parameters and some inferential tools are discussed in Section 6. In Section 7, we propose the loggamma extended Weibull regression model, which can be applied for lifetime analysis. The importance of the new models is shown empirically by means of two real data sets in Section 8. Some conclusions are offered in Section 9.
The GEW distribution
has the GEW(τ,a) distribution. This generating scheme is straightforward because of the existence of fast generators for gamma random variables.
Useful expansions
respectively. The properties of some exponentiated distributions have been investigated by several authors, see Mudholkar and Srivastava (1993) and Mudholkar et al. (1995) for exponentiated Weibull (ExpW), Gupta et al. (1998) for exponentiated Pareto, Gupta and Kundu (2001) for exponentiated exponential (ExpE) and Nadarajah and Gupta (2007) for exponentiated gamma (ExpG) distributions. More recently, Cordeiro et al. (2011) studied these properties for the exponentiated generalized gamma (ExpGG) distribution.
where \(h_{a+k}(x)=(a+k)\,\lambda \,\beta \,x^{\beta 1}\,\left (1+\alpha \lambda \,x^{\beta }\right)^{1/\alpha 1}\,\left [1\left (1+\alpha \lambda \,x^{\beta }\right)^{\frac {1}{\alpha }}\right ]^{a+k1}\) denotes the EEW(τ,a+k) density function. So, the GEW density function is a linear mixture of EEW densities.
Similarly, we can derive an expansion for the GW density when α=0. Using Eq. (5) for α=0, we obtain the same expression in (8), but the function h _{ a+k }(x) denotes, when α=0, the GEW(τ,a+k) pdf given by the second equation of (7).
where \(e_{j}= \sum ^{\infty }_{k=0}(1)^{j}\,(a+k)\,b_{k}\,\binom {a+k1}{j}\), \(g_{\lambda ^{\star },\alpha ^{\star },\beta }(x)\phantom {\dot {i}\!}\) denotes the EW pdf with parameters λ ^{⋆}=(j+1)λ, α ^{∗}=α/(j+1) and β and \(g_{\lambda ^{\star },\beta }(x)\phantom {\dot {i}\!}\) denotes the Weibull pdf with parameters λ ^{⋆} and β.
Equation (9) is the main result of this section. It reveals that the GEW density function is a linear mixture of EW (when α>0) and Weibull (when α=0) densities. So, several GEW structural properties can be obtained from those properties of the EW and Weibull distributions.
Quantile function
where a _{0}=0, a _{1}=Γ(a+1)^{1/a }, a _{2}=Γ(a+1)^{2/a }/(a+1), etc. Quantities of interest can be obtained from (10) by substituting appropriate values for u.
Moments and mean deviations
where \(e_{j}^{\star }=(j+1)\,e_{j}\), B(a,b)=Γ(a)Γ(b)/Γ(a+b) is the beta function and \(\Gamma (a)=\int _{0}^{\infty }\,z^{a1}{\mathrm {e}}^{z}dz\) is the gamma function.
We obtain the incomplete moments by combining (13) and the expression for ρ(y;r,p,q).
Estimation
The expression for l(θ) when α=0 gives the loglikelihood function for the GG distribution, a submodel of the GEW distribution. Next, we maximize \(l({\boldsymbol {\theta }})=\sum _{i=1}^{n} l_{i}({\boldsymbol {\theta }})\) for α>0.
where ψ(·) is the digamma function.
Setting these equations to zero and solving them simultaneously yield the MLEs of the four parameters. They can be solved numerically by using the Rlanguage or any iterative methods such as the NR (NewtonRaphson), BFGS (BroydenFletcherGoldfarbShanno), BHHH (BerndtHallHallHausman), NM (NelderMead), SANN (SimulatedAnnealing) and LimitedMemory QuasiNewton code for BoundConstrained Optimization (LBFGSB). Regarding the initial values for the parameters, several authors (Ugray et al. 2007; Glover 1998; Dixon and Szegö 1978; Varadhan and Gilbert 2009) suggest to find local maxima starting from widely varying starting points and then pick the maximum out of these, i.e., they suggest the use of multiple starting points (commonly referred to as multistart optimization). We followed their advice in this paper. As a useful tool we suggest the use of the R rotine multiStart of R package BB (for solving and optimizing largescale nonlinear systems) for dealing with multiple starting points to obtain multiple solutions and to test sensitivity to starting values.
For interval estimation of the model parameters, we require the 4×4 total observed information matrix J(θ). The elements of J(θ)={J _{ rs }}, where r,s∈{α,β,λ,a}, can be obtained from the authors upon request. The multivariate normal \(N_{4}\left (0,J(\widehat {\boldsymbol {\theta }})^{1}\right)\) distribution, where \(J(\widehat {\boldsymbol {\theta }})^{1}\) is the inverse observed matrix evaluated at \({\boldsymbol {\theta }}=\widehat {\boldsymbol {\theta }}\), can be used to construct approximate confidence intervals for the parameters.
The likelihood ratio (LR) statistic is useful for comparing the GEW distribution with some of its special models. We can evaluate the maximum values of the unrestricted and restricted loglikelihoods to obtain LR statistics for testing some of its submodels. In any case, hypothesis tests of the type H _{0}: ψ=ψ _{0} versus H: ψ≠ψ _{0}, where ψ is a vector formed with some components of θ and ψ _{0} is a specified vector, can be performed using LR statistics.
The LGEW regression model
The special case α=0 and a=1 refers to the logWeibull (LW) (or extremevalue) distribution and, for α=0, we obtain the loggammaWeibull (LGW) model.
In many practical applications, the lifetimes x _{ i } are affected by explanatory variables such as the cholesterol level, blood pressure and many others. Let v _{ i }=(v _{ i1},…,v _{ ip })^{ T } be the explanatory variable vector associated with the ith response variable y _{ i } for i=1,…,n. Consider a sample (y _{1},v _{1}),…,(y _{ n },v _{ n }) of n independent observations, where each random response is defined by \(y_{i}=\min \{\log (x_{i}),\log (c_{i})\}\), and \(\log (x_{i})\) and \(\log (c_{i})\) are the loglifetime and logcensoring, respectively. We consider noninformative censoring such that the observed lifetimes and censoring times are independent.
where the random error z _{ i } has the density function (15), \({\boldsymbol {\beta }}=\left (\beta _{1},\ldots,\beta _{p}\right)^{T}\), σ>0, a>0 and α>0 are unknown scalar parameters and v _{ i } is the vector of explanatory variables modeling the location parameter \(\mu _{i}=\textbf {v}_{i}^{T} {\boldsymbol {\beta }}\). Hence, the location parameter vector \({\boldsymbol {\mu }}=\left (\mu _{1},\ldots,\mu _{n}\right)^{T}\) of the LGEW model has a linear structure μ=v ^{ T } β, where \(\textbf {v}=\left (\textbf {v}_{1},\ldots,\textbf {v}_{n}\right)^{T}\) is a known model matrix. The logWeibull (or the extreme value) regression model is defined by (16) with α=0 and a=1.
where q is the observed number of failures and \(z_{i}=\left (y_{i}\textbf {v}_{i}^{T} {\boldsymbol {\beta }}\right)/\sigma \). The MLE \(\widehat {{\boldsymbol {\theta }}}\) of θ can be obtained by maximizing the loglikelihood function (17). We use the procedure NLMixed in SAS to evaluate the estimate \(\widehat {{\boldsymbol {\theta }}}\). Initial values for β and σ are taken from the fit of the LW regression model with α=0 and a=1.
Applications
8.1 The GEW model
In the first application, we use the warp break data for six types of weaving warps discussed by Tippett (1950, p. 105). We describe his experiment: “The results of a weaving experiment was conducted in a factory. There were 6 lots of warp yarn labelled respectively AL, AM, etc. They were spun from two growths of cotton, A and B, and each cotton was spun to three twists (i.e., the number of turns in the yarn per inch): low (L), medium (M), and high (H). The combination of these three factors give 6 kinds of yarn, which are the experimental treatments. From each yarn were prepared 9 warps (a warp is a quantity of warp yarn that goes into one loom as a unit), and, as a loom came available in the course of events, a warp chosen at random from the 54 was assigned to it, until ultimately all 54 were disposed of. More than one warp was woven in some looms, but that did not affect the randomness of the distribution. The number of warp threads that broke during the weaving of each warp was counted and expressed as a rate of so many breaks per unit of warp.”
We analyze the warp breakage rates for individual warps disregarding the factors. First, we fit the GEW model and some of its submodels to these data by the method of maximum likelihood. Afterwards, we compare the GEW model with some fourparameter competitive models.
The GEW model, its submodels and some competitors
MLEs of the model parameters for the warp breakage rate data, the corresponding SEs (given in parentheses) and the AIC measure
Model  α  λ  β  a  AIC 

GEW  0.895137  0.000026  9.792369  23.609396  420.257020 
(0.042691)  (0.000011)  (0.714861)  (2.615331)  
GW  0  0.008219  1.622223  2.061746  425.080529 
()  (0.000971)  (0.025583)  (0.071981)  
EW  0.104458  0.000192  2.499730  1  427.270840 
(0.025452)  (0.000032)  (0.052776)  ()  
GEE  0.000961  0.190944  1  5.364537  421.924911 
(0.013883)  (0.026556)  ()  (0.502296)  
GE  0  0.250448  1  6.896945  422.114535 
()  (0.006646)  ()  (0.176429)  
Model  k  c  θ  s  AIC 
BXIIG  1.430692  3.850236  0.885467  49.00841  422.653656 
(0.530019)  (0.094952)  (0.055033)  (11.093854)  
Model  μ  σ  a  b  AIC 
BLN  1.290503  1.322159  30.985635  2.704164  420.316246 
(0.668315)  (0.373892)  (21.297427)  (1.327947)  
Model  α  θ  β  γ  AIC 
EWP  14.50818  0.90275  0.13294  0.84268  420.51673 
(2.69087)  (0.27133)  (0.02132)  (0.04587) 
LR tests for the warp breakage rate data
Model  Hypotheses  Statistic LR  pvalue 

GEW vs GW  H _{0}:α=0 vs H _{1}:H _{0} is false  6.823509  0.008996563 
GEW vs EW  H _{0}:a=1 vs H _{1}:H _{0} is false  9.013820  0.002679457 
GEW vs GEE  H _{0}:β=1 vs H _{1}:H _{0} is false  3.667891  0.055470343 
GEW vs GE  H _{0}:α=0,β=1 vs H _{1}:H _{0} is false  5.857515  0.053463433 
The required numerical evaluations are implemented by using an R script (subroutine nlminb that can be found at https://cran.rproject.org ). The nlminb() is a derivativefree method for function minimization as it requires only the function to be evaluated. Other R routine choices such as the optimx package (Nash and Varadhan, 2011) provided similar results. The R rotine multiStart of R package BB is also used in this work. The data set warpbreaks is available in the R data frame.
Some estimation issues
This fact implies that the MLE \(\hat {{\boldsymbol {\theta }}}\) of θ=(α,λ,β,a) of the GEW model may not be unique.
As mentioned before, in order to handle the case of multiple solutions we adopt different initial points, say k, in the maximization procedure and use the estimate which maximizes the loglikelihood among these k values. For the current data sets, we have always been able to find a solution.
As mentioned above, the GEW distribution has different expressions depending on whether α>0 or α=0. In order to study possible estimation difficulties for this model when the true value of α is either zero or around it, we simulate data from the gamma generalized distribution, i.e., by taking α=0 for some choices of the parameters a, λ and β and then use the GEW model in the estimation procedure. We note that the GEW model is able to estimate all the parameter values correctly. As expected, for larger samples, the estimated values tend to be closer to the true values. Of course, for any similar problem, the program may fail to converge for some choices of initial values.
We also perform some additional analyses in order to evaluate the GEW robustness. We simulate two different data sets from the betaWeibull distribution (which does not belong to the GEW family) and use them to fit the GEW distribution. The resulting fits (not shown here) indicate that the GEW distribution is very robust. In fact, the estimated GEW densities capture the main aspects of the betaWeibull generated data in the sense that they are able to correctly locate the mode of the histograms obtained from the generated data, inflection points and other data characteristics.
Regarding the impact of the sample size on the estimated values of the parameters, a simulation study is conducted in which we generate GEW data considering (α,λ,β,a)=(0.895137,0.000026,9.792369,23.609396) as the true parameter values. We simulate data by taking k=500 replicates for each of the sample sizes n=50,100,500,1000,5000 and n=10,000. For a given sample size and k=500 estimated values we evaluate the average of those estimated vectors and the mean squared errors (MSEs). We can conclude that the estimated expected vector does approach the true vector, but the MSE decreases slowly.
8.2 The LGEW model
We now illustrate some of the ideas and methodology of regression models for the LGEW model using the data set from a twoarm clinical trial considered earlier by Efron (1988) and Mudholkar et al. (1996). Efron observed that the empirical hazard functions for both samples start near zero, suggesting an initial highrisk period in the beginning, a decline for a while, and then stabilization after about one year. Specifically, Efron’s data from a head and neck cancer clinical trial consist of survival times of 51 patients in arm A who were given radiation therapy and 45 patients in arm B who were given radiation plus chemotherapy. Nine patients in arm A and 14 patients in arm B were lost to followup and were regarded as censored. In this paper, we consider only one predictor: (v1): twoArm (Arm A=0, Arm B=1).
where the errors z _{1},…,z _{96} are independent random variables with density function (15).

LBW distribution$$\begin{array}{@{}rcl@{}} f(y)=\frac{1}{\sigma B(a,b)}\exp\left\{\left(\frac{y\mu}{\sigma}\right)b\exp\left(\frac{y\mu}{\sigma}\right) \right\} \left\{1\exp\left[\exp\left(\frac{y\mu}{\sigma}\right)\right]\right\}^{a1}, \end{array} $$
where \(\infty <y<\infty \), σ>0 and \(\infty <\mu <\infty \). See, for example, more details and properties in Cordeiro et al. (2013) and Ortega et al. (2015).

KwL distribution$$\begin{array}{@{}rcl@{}} f(y)&=&\frac{a\,b}{\sigma}\exp\left[a\left(\frac{y\mu}{\sigma}\right)\right] \left[1+\exp\left(\frac{y\mu}{\sigma}\right)\right]^{(a+1)} \\&&\times\left\{1\left[1\frac{1}{1+\exp\left(\frac{y\mu}{\sigma}\right)}\right]^{a}\right\}^{b1}, \end{array} $$
where \(\infty <y<\infty \), σ>0 and \(\infty <\mu <\infty \). Some applications of the KwL distribution are discussed in Santana et al. (2012) and Nadarajah et al. (2012).
Taking μ=β _{0}+β _{1} v _{ i1} for the location parameter, σ is the dispersion parameter and a and b are shape parameters.
MLEs of the parameters from some fitted regression models to the Efron’s data, the corresponding SEs (given in parentheses), pvalues in [.] and the AIC and BIC measures
Model  a  α  σ  β _{0}  β _{1}  AIC  BIC 

LGEW  0.3101  32.5577  0.1420  5.6000  −0.0868  281.4  294.2 
(0.0904)  (21.3182)  (0.0405)  (0.1706)  (0.1654)  
[ <0.0001]  [0.6010]  
LGW  0.0264  0  0.0510  8.0113  −0.6274  339.7  349.9 
(0.0079)  (0.0152)  (0.1458)  (0.1608)  
[ <0.0001]  [0.0002]  
LW  1  0  1.1800  6.7873  −0.7586  312.6  320.3 
(0.1082)  (0.2088)  (0.2803)  
[ <0.0001]  [0.0089]  
a  b  σ  β _{0}  β _{1}  AIC  BIC  
LBW  167.06  14.0059  12.8338  −6.0419  −0.5836  299.4  312.2 
(0.3987)  (0.0682)  (1.1065)  (1.0492)  (0.2726)  
[ <0.0001]  [0.0348]  
KwL  16.2819  312.91  5.8106  1.6912  −0.6834  308.0  320.8 
(2.6189)  (24.86)  (0.3258)  (6.0235)  (0.2792)  
[0.7795]  [0.0162] 
LR tests for the Efron’s data
Model  Hypotheses  Statistic LR  pvalue 

LGEW vs LGW  H _{0}: α=0 vs H _{1}:H _{0} is false  60.3  <0.0001 
LGEW vs LW  H _{0}: (a,α)^{ T }=(1,0)^{ T } vs H _{1}:H _{0} is false  35.2  <0.0001 
The LGEW model involves an extra parameter, which gives it more flexibility to fit the data. We note from the fitted LGEW regression model that the dummy variable v _{1} is not significant at 5 %. We note that there is no significant difference between “Arm A” and “Arm B” clinical trial for the survival times. The LGEW regression model outperforms the other models irrespective of the criteria and it can be used effectively in the analysis of these data.
Conclusions
We introduce a new model named the gamma extended Weibull (GEW) distribution and study some of its structural properties. It generalizes some important distributions in the literature and provides means of its continuous extension to still more complex situations. The new model contains several distributions as special models including the extended Weibull (Mudholkar et al. 1996), gamma Weibull (Zografos and Balakrishnan 2009) and generalized gamma (Stacy 1962). We derive explicit expressions for the density function, ordinary and incomplete moments, quantile function and mean deviations. The model parameters are estimated by maximum likelihood. The usefulness of the GEW distribution is illustrated by means of an application to real data, where we show empirically that it gives a better fit than some of its submodels. We also propose the loggamma extended Weibull (LGEW) regression model, which has greater flexibility as shown by means of an application to real data.
Declarations
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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