Compounding of distributions: a survey and new generalized classes
 Muhammad H Tahir^{1}Email authorView ORCID ID profile and
 Gauss M. Cordeiro^{2}
https://doi.org/10.1186/s4048801600521
© The Author(s) 2016
Received: 18 April 2016
Accepted: 20 September 2016
Published: 6 October 2016
Abstract
Generalizing distributions is an old practice and has ever been considered as precious as many other practical problems in statistics. It simply started with defining different mathematical functional forms, and then induction of location, scale or inequality parameters. The generalization through induction of shape parameter(s) started in 1997, and the last two decades were full of such practices. But to cope with the complex situations under series and parallel structures, the art of mixing discrete and continuous started in 1998. In this article, we present a survey on compounding univariate distributions, their extensions and classes. We review several available compound classes and propose some new ones. The recent trends in the construction of generalized and compounding classes are discussed, and the need for future works are addressed.
Keywords
AMS Subject Classification
Introduction
The modern era on distribution theory stresses on problemsolving faced by the practitioners and applied researchers and proposes a variety of models so that lifetime data set can be better assessed and investigated in different fields. In other words, there is strong need to introduce useful models for better exploration of the reallife phenomenons. Nowadays, the trends and practices in defining probability models totally differ in comparison to the models suggested before 1997. One main objective for proposing, extending or generalizing (models or their classes) is to explain how the lifetime phenomenon arises in fields like physics, computer science, insurance, public health, medical, engineering, biology, industry, communications, lifetesting and many others. For example, the wellknown and fundamental distributions such as exponential, Rayleigh, Weibull and gamma are very limited in their characteristics and are unable to show wide flexibility. The number of shape parameters, closedform expressions of the cumulative distribution function (cdf), forms of the quantile function (qf), density and failure rate shapes, skewness and kurtosis features, entropy measures, estimation of stress reliability \(R=\mathbb {P}(Y<X)\), structural properties, use of advanced mathematical functions and power series, which are available through computing platforms like Mathematica, Maple, Matlab, Python, Ox and R, variety of estimation methods, estimation of parameters in case of censored and uncensored situations, simulation results, coping with data sets having different shapes and goodnessoffits are some wellestablished characteristics that a proposed lifetime model may possess.
For complex phenomenons in reliability studies, lifetime testing, human mortality studies, engineering modeling, electronic sciences and biological surveys, the failure rate behavior can be bathtub, upsidedown bathtub and other shaped but not usually monotone increasing or decreasing. Thus, in order to cope with both monotonic and nonmonotonic failure rate shapes, researchers have introduced several generalizations (or Gclasses) which are very flexible to study needful properties of the model and its fitness.
In the last two decades, two main generalization approaches were adopted and practiced, and have also received increased attention.
1.1 First generalization approach (Gclasses)
The first approach deals with the shape parameter(s) induction in parent (or baseline) distribution to explore tail properties and to improve goodnessoffits. Some wellknown generalized (or G) classes are: MarshallOlkinG (Marshall and Olkin 1997), exponentiatedG (Gupta et al. 1998), betaG (Eugene et al. 2002), KumaraswamyG (Cordeiro and deCastro 2011), McDonaldG (Alexander et al. 2012), ZBgammaG (Zografos and Balakrishnan 2009; Amini et al. 2014), RBgammaG (Ristić and Balakrishanan 2012; Amini et al. 2014), oddgammaG (Torabi and Montazari 2012), KummerbetaG (Pescim et al. 2012), beta extended WeibullG (Cordeiro et al. 2012b), odd exponentiated generalizedG (Cordeiro et al. 2013a), truncated exponentialG (BarretoSouza and Simas 2013), logisticG (Torabi and Montazari 2014), gamma extended WeibullG (Nascimento et al. 2014), odd WeibullG (Bourguignon et al. 2014a), exponentiatedhalflogisticG (Cordeiro et al. 2014a), LibbyNovick betaG (Cordeiro et al. 2014b; Ristić et al. 2015), LomaxG (Cordeiro et al. 2014d), HarrisG (Batsidis and Lemonte 2015; Pinho et al. 2015), modified betaG (Nadarajah et al. 2014b), odd generalizedexponentialG (Tahir et al. 2015), Kumaraswamy odd loglogisticG (Alizadeh et al. 2015b), beta odd loglogisticG (Cordeiro et al. 2016), KumaraswamyMarshallOlkinG (Alizadeh et al. 2015c), betaMarshallOlkinG (Alizadeh et al. 2015a), WeibullG (Tahir et al. 2016b), exponentiatedKumaraswamyG (daSilva et al. 2016), ZBgammaoddloglogisticG (Cordeiro et al. 2015a) and Tukey’s g and hG (Jiménez et al. 2015). For more details on some wellestablished Gclasses, the reader is referred to Tahir and Nadarajah (2015).
The mathematical properties of the KumaraswamyG family were studied by Nadarajah et al. (2012), Hussian (2013), and Cordeiro and Bager (2015). The failure rate and aging properties of the betaG and ZBgammaG models were addressed by Triantafyllou and Koutras (2014). The structural properties of the ZBgammaG and RBgammaG models were recently investigated by Nadarajah et al. (2015b), and Cordeiro and Bourguignon (2016).
The revolutionary idea in generalization begun with the work of Alzaatreh et al. (2013) who proposed transformedtransformer (TX) (WeibullX and gammaX) family of distributions. This approach was further extended to the exponentiated TX (Alzaghal et al. 2013), TX{Y}quantile based approach (Aljarrah et al. 2014), TR{Y} (Alzaatreh et al. 2014), TWeibull{Y} (Almheidat et al. 2015), Tgamma{Y} (Alzaatreh et al. 2016a), TCauchy{Y}(Alzaatreh et al. 2016b), GumbelX (AlAqtash 2013; AlAqtash et al. 2014) and logisticX (Tahir et al. 2016).
After the widespread popularity of wellestablished exponentiatedG, MarshallOlkinG, betaG, KumaraswamyG and McDonaldG classes, and TX family, the transmutedG class of distributions has received increased attention in the last decade. This class is based on the quadratic rank transmutation map (QRTM) pioneered by Shaw and Buckley (2009) and highlighted by Aryal and Tsokos (2009, 2011).
where η is the transmuted (or shape) parameter. For η=0, the above equation reduces to the baseline distribution.
The general properties of the transmuted family were obtained by Bourguignon et al. (2016a) and Das (2015). The transmuted family was further extended as the exponentiated transmutedG type 1 using the Lehmann alternative type 1 (LA1) class (due to Gupta et al. 1998) by Nofal et al. (2016) and Alizadeh et al. (2016a), the exponentiated transmutedG type 2 using the Lehmann alternative type 2 (LA2) class (due to Gupta et al. 1998) by Merovci et al. (2016), and the transmuted exponentiated generalizedG by Yousof et al. (2015).
Contributed work on transmuted distributions
S.No.  Pioneer Year  Distribution  Author(s) 

1.  2009  Transmuted extreme value  Aryal and Tsokos (2009) 
2.  2011  Transmuted Weibull  Aryal and Tsokos (2011) 
Ahmad et al. (2015b)  
Khan et al. (2016b)  
3.  2013  Transmuted loglogistic  Aryal (2013) 
Granzotto and Louzada (2015)  
4.  2013  Transmuted Rayleigh  Merovci (2013a) 
Ahmad et al. (2015a)  
5.  2013  Transmuted exponentiatedexponential  Merovci (2013b) 
Khan et al. (2015a)  
6.  2013  Transmuted Fréchet  Mahmoud and Mandouh (2013) 
7.  2013  Transmuted Lomax  Ashour and Eltehiwy (2013a) 
8.  2013  Transmuted Lindley  Merovci (2013c) 
9.  2013  Transmuted quasiLindley  Elbatal and Elgarhy (2013) 
10.  2013  Transmuted exponentiatedLomax  Ashour and Eltehiwy (2013b) 
11.  2013  Transmuted modified inverse Weibull  Elbatal (2013a) 
12.  2013  Transmuted generalized inverted exponential  Elbatal (2013b) 
13.  2013  Transmuted exponentiatedmodified Weibull  Ashour and Eltehiwy (2013c) 
14.  2013  Transmuted generalized linear exponential  Elbatal et al. (2013) 
15.  2013  Transmuted additive Weibull (AW)  Elbatal and Aryal (2013) 
16.  2013  Transmuted modified Weibull (MW)  Khan and King (2013) 
17.  2014  Transmuted Pareto  Merovci and Puka (2014) 
18.  2014  Transmuted Maxwell  Iriarte and Astorga (2014) 
19.  2014  Transmuted linear exponential  Tian et al. (2014) 
20.  2014  Transmuted inverse Rayleigh (IR)  Sharma et al. (2014) 
Ahmad et al. (2014)  
21.  2014  Transmuted generalized Rayleigh (GR)  Merovci (2014) 
Iriarte and Astorga (2015)  
22.  2014  Transmuted inverted Weibull (IW)  Khan et al. (2014) 
Khan and King (2014a)  
23.  2014  Transmuted generalized IW  Merovci et al. (2014) 
Khan and King (2014b)  
24.  2014  Exponentiated transmuted Weibull  Ebraheim (2014) 
25.  2014  Beta transmuted Weibull  Pal and Tiensuwan (2014) 
26.  2014  Transmuted exponentiatedgamma  Hussian (2014) 
Lucena et al. (2015)  
27.  2014  Transmuted exponentiatedFréchet  Elbatal et al. (2014) 
28.  2015  Transmuted exponential  Owoloko et al. (2015) 
29.  2015  Transmuted Burr III  AbdulMoniem (2015) 
30.  2015  Transmuted Gompertz  AbdulMoniem and Seham (2015) 
Khan et al. (2016c)  
31.  2015  Transmuted modified IR  Khan and King (2015) 
32.  2015  Generalized transmuted AW  Mansour et al. (2015a) 
33.  2015  Kumaraswamy transmuted MW  Mansour et al. (2015b) 
34.  2015  Transmuted generalized MW  Cordeiro et al. (2015c) 
35.  2015  Kumaraswamy transmuted exponentiated MW  AlBabtain et al. (2015) 
36.  2015  Transmuted MarshallOlkin (MO) Fréchet  Afify et al. (2015a) 
37.  2015  Exponentiated transmuted GR  Afify et al. (2015b) 
38.  2015  Transmuted WeibullLomax  Afify et al. (2015c) 
39.  2015  Transmuted Dagum  Elbatal and Aryal (2015) 
Shahzad and Asghar (2016)  
40.  2015  Transmuted exponentiatedPareto  Luguterah and Nasiru (2015) 
Fatima and Roohi (2015)  
41.  2015  Generalized transmuted Lindley  Mansour and Mohamed (2015) 
42.  2015  Transmuted Chen  Khan et al. (2015b) 
43.  2015  Beta transmuted Fréchet  daSilva et al. (2015a) 
44.  2016  Transmuted Kumaraswamy  Khan et al. (2016a) 
45.  2016  Transmuted generalized Lindley  Elgarhy et al. (2016) 
46.  2016  Transmuted AlmalkiYuan MW  Vardhan and Balaswamy (2016) 
47.  2016  Transmuted power function  Haq et al. (2016) 
48.  2016  Transmuted geometric  Chakraborty and Bhati (2016) 
49.  2016  Transmuted BirnbaumSaunders  Bourguignon et al. (2016b) 
50.  2016  Kumaraswamy Transmuted MO Fréchet  Yousof et al. (2016) 
51.  2016  Transmuted WeibullPareto  Afify et al. (2016) 
1.2 Second generalization approach (compounding)
The second approach deals with the compounding of discrete models, namely the geometric, Poisson, logarithmic, binomial, negativebinomial (NB), ConwayMaxwellPoisson (COMP) and powerseries with continuous lifetime models. The basic idea of introducing compound models or families is that a lifetime of a system with N (discrete random variable) components and the positive continuous random variable, say Y _{ i } (the lifetime of the ith component), can be denoted by the nonnegative random variable Y= min(Y _{1},…,Y _{ N }) (the minimum of an unknown number of any continuous random variables) or Z= max(Y _{1},…,Y _{ N }) (the maximum of an unknown number of any continuous random variables), based on whether the components are in series or in parallel structure. Some useful references for the readers are Louzada et al. (2012a), Leahu et al. (2013) and Bidram and Alavi (2014).
The objectives of our article are threefold: Firstly, we present an uptodate account on the compounded distributions and their generalizations for the readers of modern distribution theory. Secondly, this survey will motivate the researchers to fill the gap and to furnish their work in remaining applied areas. Thirdly, this survey will also be helpful as a tutorial to the beginners of compound modeling art.
The rest of the article is organized as follows. In Section 2, two compound Gclasses are reviewed to represent the distributions of the minimum and maximum of an unknown number of continuous random variables having the same parent lifetime distribution. In Section 3, fourteen existing and new compound classes for the minimum constructed from the zerotruncated geometric (ZTG), zerotruncated Poisson (ZTP), logarithmic (Ln), zerotruncated binomial (ZTBi) and zerotruncated negativebinomial (ZTNB) distributions are described. In Section 4, we obtain the dual models for the maximum corresponding to those models discussed in Section 3. Section 5 deals with several compounding models and their extensions, which can be derived under the construction of the minimum and maximum of random variables. Sections 7 and 8 deal with other or different types of compounded models. In Section 8, we present recent trends on compounding of distributions, their Gclasses and mixing of compounded and transmuted distributions. The main purpose of Section 9 is to briefly review general inference procedure, crude rate survival models and their inference. The paper is concluded with some remarks in Section 10.
Construction of compound Gclasses
Suppose that a system has N subsystems assumed to be independent and identically distributed (i.i.d.) at a given time, where the lifetime of the ith subsystem is denoted by Y _{ i }, and that each subsystem is made of α parallel units, so that the system will fail if all the subsystems fail. We note that for a parallel system, the system fails only if all the subsystems fail, but for a series system, the failure of any subsystem will destroy the whole system. Further, suppose that the random variable N follows any discrete distribution with probability mass function (pmf) \(\mathbb {P}(N=n)\) (for n=1,2,…). We consider that the failure times of the components Z _{ i,1},…,Z _{ i,α } for the ith subsystem are i.i.d. random variables having a suitable cdf, which is a function of the baseline cdf depending on a parameter vector τ, say T[G(x;τ),α]=G(x;τ)^{ α } (for x>0). In the following construction, although α is a positive integer called power or resilience parameter, we can consider that α>0.
Many compound Gclasses can be constructed from Eqs. (4) and (5) by choosing a discrete model with pmf \(\mathbb {P}(N=n)\). The random variables Y= min{Y _{1},…,Y _{ N }} and Z= max{Y _{1},…,Y _{ N }} generate several models that arise in series and parallel systems with identical components and have many industrial and biological applications. For example, the time to the failure of a series system with an unknown number of protected components or the cancer recurrence of an individual after undergoing a certain treatment can be modeled by the generated distribution of Y. In a dual mechanism, the time to the failure of a parallel system with an unknown number of protected components or a disease manifestation, if it occurs only after an unknown number of factors have been active, can be modeled by the generated distribution of Z.
Compound Gclasses
In this section, we present 14 compounded models obtained from Eq. (4). In Section 4, we present the corresponding complementary models based on Eq. (5). The list below does not include all compounded models but a large number of them and some new ones. For example, it does not describe the exponentiatedGConwayMaxwell Poisson (EGCOMP) class pioneered by Cordeiro et al. (2012a) and its special GCOMP model, among others. For all formulated models, we provide only their cdfs since the corresponding probability density functions (pdfs) can be determined by simple differentiation.
3.1 Exponentiated Ggeometric class
The EGG class has recently been introduced by Nadarajah et al. (2015a), and can also be called the generalized Ggeometric class. For α=1, the EGG class reduces to the Ggeometric (GG) class proposed by Alkarni (2012). They investigated some of its general properties.
Remark 1
For α=1, the EGGA class reduces to the alternative Ggeometric (GGA) class defined by Castellares and Lemonte ( 2016 ).
3.2 Exponentiated KumaraswamyGgeometric class (new)
For α=1, the EKGG class gives the new KumaraswamyG geometric (KGG) family.
3.3 McDonaldGgeometric class (new)
Alexander et al. (2012) defined the cdf of the McDonaldG class by \(\phantom {\dot {i}\!}M_{a,b,c}(x;\tau)=I_{G(x;\tau)^{c}}(a,b)\), where \(\phantom {\dot {i}\!}B(a,b)={\int _{0}^{1}} w^{a1}\, (1w)^{b1}\, dw\), \(\phantom {\dot {i}\!}B_{z}(a,b)={\int _{0}^{z}} w^{a1}\, (1w)^{b1}\, dw\) and I _{ z }(a,b)=B _{ z }(a,b)/B(a,b) are the beta function, incomplete beta function and incomplete beta function ratio, respectively.
3.4 BetaGgeometric class (new)
3.5 Exponentiated GPoisson class
The EGP class has been studied by Gomes et al. (2015). For α=1, it becomes the GPoisson (GP) class as defined recently by Tahir et al. (2016a) by the name of the PoissonX class since it was based on the TX family.
3.6 Exponentiated KumaraswamyGPoisson class (new)
For α=1, the EKGP class reduces to the KumaraswamyG Poisson (KGP) family studied by Ramos et al. (2015).
3.7 McDonaldGPoisson class (new)
3.8 BetaGPoisson class (new)
3.9 Exponentiated Glogarithmic class (new)
by noting that \(\sum _{n=1}^{\infty }\,Q^{n}/n=\,\ln (1Q)\).
For α=1, the EGLn class becomes the Glogarithmic (GLn) family introduced by Alkarni (2012).
3.10 Exponentiated KumaraswamyGlogarithmic class (new)
For α=1, the EKGLn cdf is identical to the cdf of the new KumaraswamyGlogarithmic (KGLn) class.
3.11 McDonaldGlogarithmic class (new)
3.12 BetaGlogarithmic class (new)
3.13 Exponentiated Gbinomial class (new)
Bakouch et al. (2012b) studied a special case of the EGBi class socalled the exponentiatedexponential binomial (EEBi) model. For α=1, the EGBi class becomes the Gbinomial (GBi) class pioneered by Alkarni (2013).
3.14 Exponentiated GNB class (new)
For α=1, the EGNB class leads to the GNB (GNB) class proposed by Percontini et al. (2013b).
Remark 2

If we replace the probability of success β by 1−β and the dispersion parameter s by θ in (6), the cdf of an alternate form of the GNB class (denoted by EGNB1) will be$$\begin{array}{@{}rcl@{}} F_{EGNB1}(y)=1\frac{\beta^{\theta}}{1\beta^{\theta}}\, \left\{\left[G(y;\tau)^{\alpha}+(1\beta)\bar{G}(y;\tau)^{\alpha}\right]^{\theta}1\right\}. \end{array} $$(7)
For α=1, the EGNB1 class leads to the GNB family discussed by Nadarajah et al. ( 2013b ), who also studied, as a special case of the GNB1 class, the exponentialtruncated negativebinomial (ETNB) model.

If we express β in terms of the population mean of the distribution in (6), the cdf of an alternate form of the GNB class (denoted by EGNB2) will be$$\begin{array}{@{}rcl@{}} F_{EGNB2}(y)&=&\frac{1\left[1+\eta\theta\,G(y;\tau)^{\alpha}\right]^{\frac{1}{\eta}}}{1(1+\eta\theta)^{\frac{1}{\eta}}}. \end{array} $$
For α=1, the EGNB2 class leads to the GNB family. Louzada et al. ( 2012b ) studied a special model of this class.
Complementary compound Gclasses
Complementary compound models are constructed by considering the maximum of a sequence of i.i.d. random variables which represents the risk time of a system having components in parallel structure. In this section, we generate from Eq. (5) the complementary Gclasses of those ones presented in Section 3. Some proposed complementary Gclasses are really new ones.
4.1 Complementary exponentiated Ggeometric class (new)
This equation is also called the complementary generalized Ggeometric family. For α=1, the CEGG class becomes the complementary Ggeometric (CGG) family.
Remark 3
For α=1, the CEGGA class leads to the complementary Ggeometric (CGG) family proposed by Castellares and Lemonte ( 2016 ).
4.2 Complementary exponentiated KumaraswamyGgeometric class (new)
For α=1, the CEKGG class leads to the complementary KGG (CKGG) class.
4.3 Complementary McDonaldGgeometric class (new)
4.4 Complementary betaGgeometric class (new)
4.5 Complementary exponentiated GPoisson class (new)
For α=1, the CEGP class leads to the complementary GPoisson (CGP) class.
4.6 Complementary exponentiated KumaraswamyG Poisson class (new)
For α=1, the CEKGP class reduces to the complementary KumaraswamyG Poisson (CKGP) class.
4.7 Complementary McDonaldG Poisson class (new)
4.8 Complementary beta GPoisson class (new)
4.9 Complementary exponentiated Glogarithmic class (new)
by noting that \(\sum _{n=1}^{\infty }\,Q^{n}/n=\,\ln (1Q)\). For α=1, the CEGLn class becomes the complementary Glogarithmic (CGLn) class.
4.10 Complementary exponentiated KumaraswamyGlogarithmic class (new)
For α=1, the complementary KumaraswamyGlogarithmic (CEKGLn) class becomes the complementary Glogarithmic (CGLn) class.
4.11 Complementary McDonaldGlogarithmic class (new)
4.12 Complementary betaGlogarithmic class (new)
4.13 Complementary exponentiated Gbinomial class (new)
For α=1, the CEGBi class becomes the complementary Gbinomial (CGBi) class.
4.14 Complementary exponentiated GNB class (new)
For α=1, the CEGNB class leads to the new complementary GNB class.
Review of existing compounded models
In this section, we review some available compounded models. In the literature, several authors have reported compounding discrete distributions, namely the ZTG, ZTP, logarithmic, ZTBi, ZTNB, zerotruncated generalized Poisson and zerotruncated powerseries, with continuous lifetime models.
5.1 Compounded models based on geometric distribution
For the following models, X denotes the r.v. of the baseline G model.
Adamidis et al. (2005) also defined an extended EG model.
Ristić and Kundu (2016) proposed the generalized geometric extreme distribution, which is identical to the EEG model given above.
where \(K=\frac {\Gamma {(l)}\Gamma {(m)}}{\Gamma {(l+m)}}\, {}_{1}F_{1} (l;l+m;c)\) and _{1} F _{1} is the Gauss hypergeometric function.
5.2 Compounded models based on Poisson distribution
For some structural properties and applications of the EP model, the reader is referred to Kuş (2007) and Cancho et al. (2011a).
The Burr III distribution is also known as the Dagum distribution (Dagum1977), which is a very wellknown model for studying income and wealth inequality data. Oluyede et al. (2016b) have recently introduced and studied the DagumPoisson distribution.
5.3 Compounded models based on the logarithmic distribution
The following compounded models have been reported in the literature from the logarithmic discrete model.
where γ>0 is the power parameter.
where ϕ∈(0,1), α>0 and δ>0 are shape parameters and β>0 is a scale parameter.
Other compounded models
where a _{ n } depends only on n, \(C(\theta) = \sum _{n=1}^{\infty }a_{n} \,\theta ^{n}\) and θ>0 is such that C(θ) is finite. Equation (9) summarizes some power series distributions (truncated at zero) such as the Poisson, logarithmic and geometric distributions, where C(θ) is equal to (e^{ θ }−1), − log(1−θ) and θ(1−θ)^{−1}, respectively.
Equation (10) includes as special cases the Weibull power series (WPS) class, which extends the exponential power series (EPS) family. In fact, this class includes much more than 60 (20 × 3) special models, some of them given by Silva et al. (2013) and others yet not investigated. In a similar context, more recently, Silva et al. (2016) defined a family by compounding the generalized gamma (GGa) and power series distributions.
The CEWPS class can arise in parallel systems with identical components, which appear in many industrial and biological applications.
The EGR distribution is useful for modeling the time between the first failure to the last failure.
A different approach of compounding
The compounding of some models (continuous with continuous and discrete with discrete) are introduced such as the exponentialWeibull (Cordeiro et al.2014c), generalized exponentialexponential (Popović et al.2015), geometric exponential Poisson (Nadarajah et al.2013a) and additive Weibull (Xie and Lai1995) distributions.
We consider that α>0, β>0 and γ∈(0,∞)∖{1}, which gives identifiability to the model. The mathematical properties of the EW model were investigated by Cordeiro et al. (2014c).
If α=1 and X∼E x p(β) in the EGGP class above, then it follows the cdf of the geometric exponential Poisson (GEP) defined by Nadarajah et al. (2013a).
(iv) Additive Weibull (AW) model. Suppose a system composed of two interconnected independent series subsystems that affect the system in a different way, both following the Weibull distribution with proper parameters. Xie and Lai (1995) proposed the AW model based on the simple idea of combining the failure rates of two Weibull distributions: one has a decreasing failure rate and the other one has an increasing failure rate. The cdf of the AW model is given by F(t)=1− exp(−a t ^{ b }−c t ^{ d }), where a>0 and c>0 are scale parameters and b>d>0 (or d>b>0) are shape parameters, which gives identifiability to the model. The interpretation of the AW model is evident. A stateoftheart survey on the AW model can be found in Lemonte et al. (2014).
Recent trends in compounding
There are four very recent trends on compounding of distributions, which have received a great deal of attention.
8.1 First recent trend
Compounding a Gclass with discrete model: The first recent trend deals with defining compound classes of lifetime distributions rather than studying a single compound model. In this technique, a noncompound Gfamily of distributions is compounded with a discrete model to generate a new flexible compounded class. Asgharzadeh et al. (2014) introduced the GPoissonLindley (from discrete PoissonLindley) class of distributions by compounding the ZTPL distribution with any other continuous lifetime model. Four special models viz. Weibull PoissonLindley, Burr PoissonLindley, exponentiatedWeibull PoissonLindley and Dagum PoissonLindley were investigated. Nadarajah et al. (2015a) proposed the exponentiated Ggeometric family, and reported two special models: exponentiatedWeibull geometric and exponentiatedloglogistic geometric. Ramos et al. (2015) introduced the KumaraswamyG Poisson family and showed that the special model KumaraswamyWeibull Poisson outperforms the competitors KumaraswamyWeibull and betaWeibull models in studying real life data on bladder cancer. Gomes et al. (2015) proposed the exponentiatedG Poisson family and studied two special models, namely the exponentiatedBurr XII Poisson and exponentiatedWeibull Poisson. Two other compound Gclasses are the GPoisson (Alkarni and Oraby2012) and PoissonX (Tahir et al. 2016a).
8.2 Second recent trend
Combining a continuous model with compound power series class: For the second recent trend on compounding, one continuous lifetime model is compounded with the power series class of distributions truncated at zero. Chahkandi and Gangali (2009) first suggested compounding exponential and power series class, which exhibits decreasing failure rate. The power series class can be used to construct many compounding models with discrete distributions: Poisson, logarithmic, geometric, binomial and negativebinomial. Some wellknown compound models defined from the power series class are: Weibull power series (WPS) (Morais and BarretoSouza2011), complementary generalizedexponential power series (CGEPS) (Mahmoudi and Jafari2012), complementary exponentiatedWeibull power series (CEWPS) (Mahmoudi and Shiran2012b), extended WPS (Silva et al.2013), Kumaraswamy power series (KwPS) (Bidram and Nekouhou2013), complementary exponential power series (CEPS) (Flores et al.2013), BirnbaumSaunders power series (BSPS) (Bourguignon et al.2014b), complementary WPS (Munteanu2014), complementary Erlang and Erlang power series (CErPS and ErPS) (Leahu et al.2014), complementary extended WPS (Cordeiro and Silva2014), exponentiated extended WPS (Tahmasebi and Jafari2015a), Burr XII power series (BIIPS) (Silva and Cordeiro2015), Lindley power series (LPS) (WarahenaLiyanage and Pararai2015a), linear failure rate power series (LFRPS) (Mahmoudi and Jafari2015), complementary normal power series (CNPS) (Mahmoudi and Mahmoodian2015), complementary generalized Gompertz power series (CGGoPS) (Tahmasebi and Jafari2015b), complementary inverse Weibull power series (CIWPS) (Shafiei et al.2016), complementary generalized modified Weibull (CGMW) (Bagheri et al.2016), complementary exponentiated inverse Weibull power series (CEIWPS) (Hassan et al.2016), generalized gamma power series (GGPS) (Silva et al.2016), Gompertz power series (GoPS) (Jafari and Tahmasebi2016), complementary generalized linear failure rate power series (CGLFR) (Harandi and Alamatsaz2016), and Dagum power series (DaPS) (Oluyede et al.2016b).
8.3 Third recent trend
Combining compound Gclass with the noncompound Gclass: Here, the cdf of one compound Gclass or its special model is inducted into the cdf of a noncompound Gclass, thus generating a new flexible Gclass. Next, we propose two such classes.
Pararai et al. (2015a) obtained the cdf of a special KwGP model called the Kumaraswamy LindleyPoisson (KwLP) distribution.
8.4 Fourth recent trend
Combining transmuted Gclass with wellknown compound distributions: After receiving increased attention in the last decade, more than 50 transmuted distributions have been reported in the literature. Due to wide acceptability of transmuted Gclass, a new trend has now begun by inserting the cdf of the compound Gclass or distribution in the transmuted Gclass cdf. Some models are given below:
where η and θ>0 are transmuted and scale parameters, respectively, and p∈(0,1).
where α>0, β>0 and γ∈(0,∞)∖{1}.
Estimation and inference
9.1 General estimation procedure
Several approaches for parameter estimation were proposed in the literature but the maximum likelihood method is the most commonly employed. The maximum likelihood estimators (MLEs) enjoy desirable properties and can be used when constructing confidence intervals and regions and also in test statistics. The normal approximation for these estimators in large sample distribution theory is easily handled either analytically or numerically. So, we consider the estimation of the unknown parameters for each model discussed in this paper from complete or censored samples by maximum likelihood.
The loglikelihood for the model parameters can be maximized either directly by using the R (optim function), SAS (PROC NLMIXED), Ox program (MaxBFGS subroutine) or by solving the nonlinear likelihood equations obtained by differentiating the loglikelihood. In the applications, we can also use the AdequacyModel package (version 2.0.0) available in the R programming language. It has been cited very frequently in papers related to new lifetime distributions. The package has been continuously updated and more information can be obtained from http://cran.rstudio.com/web/packages/AdequacyModel/index.html. It is distributed under the terms of the GNU licenses  GNU Project. An important observation is that it is not necessary to define the loglikelihood function but only the pdf and cdf of the model. The package provides some useful goodnessoffit statistics to assess the quality of the fitted models and compare them, such as the Cramér–von Mises (W ^{∗}) and AndersonDarling (A ^{∗}) statistics, Akaike information criterion (AIC), consistent Akaike information criterion (CAIC), Bayesian information criterion (BIC), HannanQuinn information criterion (HQIC) and KolmogorovSmirnov (KS) test. It is important to emphasize that we can fit several competitive models to a data set and select those which yield best fits by means of the AdequacyModel package.
9.2 Cure fraction survivor models
During the last two decades, the tendency to propose probability models to deal with survival data has increased. This increased interest has lead researchers and medical practitioners to assess correct causes and information of the disease. The survival models which are receiving increased recognition in these days are fractional survivor models. These models have been effectively useful in some situations of clinical, medical or biological studies, where the fractional survival (or survivor fraction) models are useful to study a cure fraction of individuals. These models are also known as cure rate models or longterm survival models. In these models, it is assumed that all units under study are susceptible to an event of interest and will eventually experience it if its followup is sufficiently long. However, there are situations for which a fraction of individuals is not expected to experience the event of interest, that is, those individuals are cured or insusceptible. For example, researchers may be interested in analyzing the recurrence of a disease but many individuals may never have an experience or a recurrence, therefore, a cured fraction of the population exist. In other words, the cure rate models cover the situations where the sampling units insusceptible to the occurrence of the event are of interest, and also extend understanding of timetoevent data by allowing the formulation of more accurate and informative conclusions. If the cure fraction of the population is ignored, then the results will match to standard survival analysis. The cure rate models have been used for modeling timetoevent data for cardiac failure and various types of cancers including prostate, breast, leukemia, nonHodgkin lymphoma and melanoma. That is why, the focus of the researchers is to introduce new, extended or modified distributions which accommodate cure fraction.
Some selected cure rate survival models
S.No.  Investigation  Cure rate model  Author(s) 

1.  Cutaneous melanoma  COMPoisson  Rodrigues et al. (2009b) 
2.  Reducing drug abuse  Generalized exponential  Kannan et al. (2010) 
3.  Malignant melanoma  Negative binomial  Cancho et al. (2011b) 
4.  Malignant melanoma  Geometric BirnbaumSaunders  Cancho et al. (2012) 
5.  Prostrate Cancer  Negative binomialbeta Weibull  Ortega et al. (2012) 
6.  Ovarian cancer  PEregression  Louzada et al. (2012c) 
7.  Myelomatosis (bone marrow)  EG  Roman et al. (2012) 
Leukemia (autologous marrow)  
8.  Gastric cancer  Generalized modified Weibull  Martinez et al. (2013) 
9.  Cutaneous melanoma  Destructive negativebinomial  Cancho et al. (2013a) 
10.  Cutaneous melanoma  Power series  Cancho et al. (2013b) 
11.  Cutaneous melanoma  COMPoisson  Balakrishnan and Pal (2013a) 
12.  Cutaneous melanoma  Negativebinomial GGa  Ortega et al. (2014) 
13.  Breast cancer  Poisson BirnbaumSaunders  Hashimoto et al. (2014) 
Hemophiliacs  
14.  Red flour beetles  logWeibull Negativebinomial  Louzada et al. (2015) 
15.  Melanoma  Weibull Negativebinomial  Yiqi et al. (2016) 
16.  Congenital malformations  Negative binomialWeibull  Hashimoto et al. (2015) 
Hemophiliacs  
17.  Breast carcinoma  Power series BW  Ortega et al. (2015) 
18.  First calving of cows  Transmuted loglogistic  Louzada and Granzotto (2015) 
19.  Cutaneous melanoma  Destructive Negativebinomial  Gallardo and Bolfarine (2016) 
20.  Malignant melanoma  Negative binomial BirnbaumSaunders  Cordeiro et al. (2016) 
9.3 Inference for cure fraction models
The EM algorithm (Dempster et al.1977) is also a very popular maximization alternative used to obtain the estimates when the model has missing data. In the literature, Balakrishnan and Pal (2012,2013b,2015a,2015b), Gallardo et al. (2016), Gallardo and Bolfarine (2016) and some others have considered estimation of parameters of cure fraction survival models using the EM algorithm.
Final remarks
The need of compounding was first felt in actuarial science and later researchers of many other fields adopted this approach for lifetime and reliability modeling. We follow the two basic principles (the minimum and the maximum) used in series and parallel structure, and report more than 30 compound Gclasses. In this way, the possible available compound Gclasses are surveyed and using these basic principles nearly 25 new Gclasses are proposed. The purpose of providing a variety of Gclasses is to test flexibility of the proposed compound models to cope with the data available in complex situations. The parameters inducted in this way might be helpful in exploring phenomenon generated from reallifetime data sets. We expect that these Gclasses or generated compounded models from them will be an addition to the art of constructing useful probability models. One can imagine its motivation and usefulness in the fields which are not touched earlier. We have also briefly described the latest trends in the development of compounding technique, which portray better exposition of the strategies adopted for the researchers and practitioners. We hope to produce many more new compound Gclasses from the function T[G(x;τ);α] but due to space problem we did our best to explore and present the elusive task in most tenable way by mentioning only the cumulative distributions of the classes. The remaining Gclasses will appear in another article under the same series. Lastly, we offer more choices to the learners and practitioners of modeling to compare different models and to study pros and cons of old and new Gclasses. The possible future projects are: (i) to propose more new compound Gclasses of distributions; (ii) to review and develop bivariate compound Gclasses; and (iii) to prepare a review and new developments on cure rate survival models.
Declarations
Acknowledgements
Both authors are very grateful to the EditorinChief Felix Famoye, an associate editor and two referees for their constructive comments which greatly improved the earlier version of our manuscript.
Authors’ contributions
The authors MHT and GMC with the consultation of each other carried out this work and drafted the manuscript together. Both authors read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
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
References
 AbdulMoniem, IB: Transmuted Burr III distribution. J. Statist.–Adv. Theory Applic. 14, 37–47 (2015).View ArticleGoogle Scholar
 AbdulMoniem, IB, Seham, M: Transmuted Gompertz distribution. Comput. Appl. Math. J. 1, 88–96 (2015).Google Scholar
 Adamidis, K, Dimitrakopoulou, T, Loukas, S: On an extension of the exponentialgeometric distribution. Statist. Probab. Lett. 73, 259–269 (2005).MathSciNetMATHView ArticleGoogle Scholar
 Adamidis, K, Loukas, S: A lifetime distribution with decreasing failure rate. Statist. Probab. Lett. 39, 35–42 (1998).MathSciNetMATHView ArticleGoogle Scholar
 Afify, AZ, Nofal, ZM, Butt, NS: Transmuted complementary Weibull geometric distribution. Pak. J. Statist. Oper. Res. 10, 435–454 (2014).MathSciNetView ArticleGoogle Scholar
 Afify, AZ, Hamedani, GG, Ghosh, I, Mead, ME: The transmuted MarshallOlkin Fréchet distribution: Properties and applications. Int. J. Statist. Probab. 4, 132–148 (2015a).Google Scholar
 Afify, AZ, Nofal, ZM, Butt, NS: Exponentiated transmuted generalized Rayleigh distribution: A new fourparameter Rayleigh distribution. Pak. J. Statist. Oper. Res. 11, 115–134 (2015b).Google Scholar
 Afify, AZ, Nofal, ZM, Yousof, HM, ElGebaly, YM, Butt, NS: The transmuted WeibullLomax distribution: Properties and application. Pak. J. Statist. Oper. Res. 11, 135–152 (2015c).Google Scholar
 Afify, AZ, Yousof, HM, Butt, NS, Hamedani, GG: The transmuted WeibullPareto distribution. Pak. J. Statist. 32, 183–206 (2016).Google Scholar
 Ahmad, A, Ahmad, SP, Ahmad, A: Transmuted inverse Rayleigh distribution: A generalization of the inverse Rayleigh distribution. Math. Theory Model. 4, 177–185 (2014).Google Scholar
 Ahmad, A, Ahmad, SP, Ahmad, A: Characterization and estimation of transmuted Rayleigh distribution. J. Statist. Applic. Probab. 4, 315–321 (2015a).Google Scholar
 Ahmad, K, Ahmad, SP, Ahmad, A: Structural properties of transmuted Weibull distribution. J. Mod. Appl. Stat. Methods. 14, 141–158 (2015b).Google Scholar
 AlAqtash, R: On Generating New Families of Distributions Using the Logit Function. Central Michigan University. Department of Mathematics, Ph.D. thesis (2013).Google Scholar
 AlAqtash, R, Lee, C, Famoye, F: GumbelWeibull distribution: Properties and applications. J. Mod. Appl. Stat. Methods. 13, 201–225 (2014).Google Scholar
 AlBabtain, A, Fattah, AA, Ahmed, AHN: The Kumaraswamytransmuted exponentiated modified Weibull distribution. Commun. Stat. Simul. Comput (2015). forthcoming.Google Scholar
 Alexander, C, Cordeiro, GM, Ortega, EMM, Sarabia, JM: Generalized betagenerated distributions. Comput. Stat. Data Anal. 56, 1880–1897 (2012).MathSciNetMATHView ArticleGoogle Scholar
 Alizadeh, M, Cordeiro, GM, deBrito, E, Demétrio, CGB: The beta MarshallOlkin family of distributions. J. Stat. Dist. Applic. 2, Art, 4 (2015a).Google Scholar
 Alizadeh, M, Emadi, M, Doostparast, M, Cordeiro, GM, Ortega, EMM, Pescim, RR: A new family of distributions: the Kumaraswamy odd loglogistic, properties and applications. Hacet. J. Math. Stat. 44, 1491–1512 (2015b).Google Scholar
 Alizadeh, M, Tahir, MH, Cordeiro, GM, Mansoor, M, Zubair, M, Hamedani, GG: The Kumaraswamy MarshalOlkin family of distributions. J. Egyptian Math. Soc. 23, 546–557 (2015c).Google Scholar
 Alizadeh, M, Bagheri, SF, Alizadeh, M, Nadarajah, S: A new fourparameter lifetime distribution. J. Appl. Statist. (2016a). doi:http://dx.doi.org/10.1080/02664763.2016.1182137. forthcoming.
 Alizadeh, M, Merovci, F, Hamedani, GG: Generalized transmuted family of distributions. Properties and applications. Hacet. J. Math. Stat. (2016b). doi:http://dx.doi.org/10.15672/HJMS.201610915478. forthcoming.
 Aljarrah, MA, Lee, C, Famoye, F: On generating TX family of distributions using quantile functions. J. Stat. Dist. Applic. 1, Art, 2 (2014).Google Scholar
 Alkarni, SH: A compound class of geometric and lifetime disributions. Open Statist. Probab. J. 5, 1–5 (2012).MathSciNetGoogle Scholar
 Alkarni, S, Oraby, A: A compound class of Poisson and lifetime disributions. J. Statist. Applic. Probab. 1, 45–51 (2012).View ArticleGoogle Scholar
 Alkarni, SH: A compound class of truncated binomial lifetime disributions. Open. J. Statist. 3, 305–311 (2013).Google Scholar
 Almheidat, M, Famoye, F, Lee, C: Some generalized families of Weibull distribution: Properties and applications. Int. J. Statist. Probab. 4, 18–35 (2015).View ArticleGoogle Scholar
 Alzaatreh, A, Lee, C, Famoye, F: A new method for generating families of continuous distributions. Metron. 71, 63–79 (2013).MathSciNetMATHView ArticleGoogle Scholar
 Alzaatreh, A, Lee, C, Famoye, F: Tnormal family of distributions: a new approach to generalize the normal distribution. J. Stat. Dist. Applic. 1, Art, 16 (2014).Google Scholar
 Alzaatreh, A, Lee, C, Famoye, F: Family of generalized gamma distributions: Properties and applications. Hacet. J. Math. Stat. 45, 869–886 (2016a).Google Scholar
 Alzaatreh, A, Lee, C, Famoye, F, Ghosh, I: The generalized Cauchy family of dstributions with applications. J. Stat. Dist. Applic. 3, Art, 12 (2016b).Google Scholar
 Alzaghal, A, Famoye, F, Lee, C: Exponentiated TX family of distributions with some applications. Int. J. Probab. Statist. 2, 31–49 (2013).View ArticleGoogle Scholar
 AlZahrani, B, Fattah, AA, Nadarajah, S, Ahmed, AHN: The exponentiated transmuted Weibull geometric distribution with application in survival analysis. Commun. Stat. Simul. Comput. (2015). forthcoming.Google Scholar
 AlZahrani, B, Sagor, H: The PoissonLomax distribution. Rev. Colombiana Estadíst. 37, 225–245 (2014).MathSciNetView ArticleGoogle Scholar
 AlZahrani, B, Sagor, H: Statistical analysis of Lomaxlogarithmic distribution. J. Stat. Comput. Simul. 85, 1883–1901 (2015).MathSciNetView ArticleGoogle Scholar
 Amini, M, MirMostafaee, SMTK, Ahmadi, J: Loggammagenerated families of distributions. Statistics. 48, 913–932 (2014).MathSciNetMATHView ArticleGoogle Scholar
 Aryal, GR: Transmuted loglogistic distribution. J. Statist. Applic. Probab. 2, 11–20 (2013).View ArticleGoogle Scholar
 Aryal, GR, Tsokos, CP: On the transmuted extreme value distribution with application. Nonlinear Anal. 71, 1401–1407 (2009).MathSciNetMATHView ArticleGoogle Scholar
 Aryal, GR, Tosokos, CP: Transmuted Weibull distribution: A generalization of Weibull probability distribution. Eur. J. Pure Appl. Math. 4, 89–102 (2011).MathSciNetGoogle Scholar
 Asgharzadeh, A, Bakouch, HS, Nadarajah, S, Esmaeili, L: A new family of compound lifetime distributions. Kybernetika. 50, 142–169 (2014).MathSciNetMATHGoogle Scholar
 Ashour, SK, Eltehiwy, MA: Transmuted Lomax distribution. Amer. J. Appl. Math. Statist. 1, 121–127 (2013a).Google Scholar
 Ashour, SK, Eltehiwy, MA: Transmuted exponentiated Lomax distribution. Austral. J. Basic Appl. Sci. 7, 658–667 (2013b).Google Scholar
 Ashour, SK, Eltehiwy, MA: Transmuted exponentiated modified Weibull distribution. Int. J. Basic Appl. Sci. 2, 258–269 (2013c).Google Scholar
 Ashour, SK, Eltehiwy, MA: Exponentiated power Lindley distribution. J. Adv. Res. 6, 895–905 (2015).View ArticleGoogle Scholar
 Ashour, SK, Wahed, MLA: Kummer betaWeibull geometric distribution: A new generalizations of betaWeibull geometric distribution. Int. J. Sci. Basic Appl. Res. 16, 258–273 (2014).Google Scholar
 Bagheri, SF, Samani, EB, Ganjali, M: The generalized modified Weibull power series distribution: Theory and applications. Comput. Stat. Data Anal. 94, 136–160 (2016).MathSciNetView ArticleGoogle Scholar
 Bakouch, HS, AlZahrani, BM, AlShomrani, AA, Marchi, VAA, Louzada, F: An extended Lindley distribution. J. Korean Stat. Soc. 41, 75–85 (2012a).Google Scholar
 Bakouch, HS, Ristić, MM, Asgharzadeh, A, Esmaily, L, AlZahrani, BM: An exponentiated exponential binomial distribution with application. Statist. Probab. Lett. 82, 1067–1081 (2012b).Google Scholar
 Balakrishnan, N, Pal, S: EM algorithmbased likelihood estimation for some cure rate models. J. Stat. Theory Prac. 6, 698–724 (2012).MathSciNetView ArticleGoogle Scholar
 Balakrishnan, N, Pal, S: Lognormal lifetimes and likelihoodbased inference for flexible cure rate models based on COMPoisson family. Comput. Stat. Data Anal. 67, 41–67 (2013a).Google Scholar
 Balakrishnan, N, Pal, S: Expectation maximizationbased likelihood inference for flexible cure rate models with Weibull lifetimes (2013b). doi:http://dx.doi.org/10.1177/0962280213.
 Balakrishnan, N, Pal, S: An EM algorithm for the estimation of parameters of a flexible cure rate model with generalized gamma lifetime and model discrimination using likelihoodand informationbased methods. Comput. Statist. 30, 151–189 (2015a).Google Scholar
 Balakrishnan, N, Pal, S: Likelihood inference for flexible cure rate models with gamma lifetimes. Commun. Stat. Theory Methods. 44, 4007–4048 (2015b).Google Scholar
 BarretoSouza, W, deMorais, AL, Cordeiro, GM: The Weibullgeometric distribution. J. Stat. Comput. Simul. 81, 645–657 (2011).MathSciNetMATHView ArticleGoogle Scholar
 BarretoSouza, B, CribariNeto, F: A generalization of the exponentialPoisson distribution. Statist. Probab. Lett. 79, 2493–2500 (2009).MathSciNetMATHView ArticleGoogle Scholar
 BarretoSouza, W, Simas, AB: The expG family of distributions. Braz. J. Probab. Statist. 27, 84–109 (2013).MathSciNetMATHView ArticleGoogle Scholar
 Batsidis, A, Lemonte, AJ: On the Harris extended family of distributions. Statistics. 49, 1400–1421 (2015).MathSciNetMATHView ArticleGoogle Scholar
 Bereta, EMP, Louzanda, F, Franco, MAP: The PoissonWeibull distribution. Adv. Applic. Statist. 22, 107–118 (2011).MathSciNetMATHGoogle Scholar
 Berkson, J, Gage, RP: Survival curve for cancer patients following treatment. J. Amer. Statist. Assoc. 47, 501–515 (1952).View ArticleGoogle Scholar
 Bidram, H: The beta exponentialgeometric distribution. Commun. Stat. Simul. Comput. 41, 1606–1622 (2012).MathSciNetMATHView ArticleGoogle Scholar
 Bidram, H, Behboodian, J, Towhidi, M: The beta Weibullgeometric distribution. J. Stat. Comput. Simul. 83, 52–67 (2013).MathSciNetMATHView ArticleGoogle Scholar
 Bidram, H, Alavi, SM: A note on exponentiated Fgeometric distributions. J. Mod. Math. Front. 3, 18–23 (2014).View ArticleGoogle Scholar
 Bidram, H, Nadadrajah, S: A new lifetime model with decreasing, increasing, bathtubshaped, and upsidedown bathtubshaped hazard rate function. Statistics. 50, 139–156 (2016).MathSciNetMATHView ArticleGoogle Scholar
 Bidram, H, Nekouhou, V: Double bounded Kumaraswamypower series class of distributions. Statist. Oper. Res. Trans. 37, 211–230 (2013).MathSciNetMATHGoogle Scholar
 Boag, JW: Maximum likelihood estimates of the proportion of patients cured by cancer therapy. J. R. Stat. Soc. B 11, 15–53 (1949).Google Scholar
 Bordbar, F, Nematollah, AR: The modified exponentialgeometric distribution. Commun. Stat. Theory Methods. 45, 173–181 (2016).MathSciNetMATHView ArticleGoogle Scholar
 Bourguignon, M, Silva, RB, Cordeiro, GM: The WeibullG family of probability distributions. J. Data Sci. 12, 53–68 (2014a).Google Scholar
 Bourguignon, M, Silva, RB, Cordeiro, GM: A new class of fatigue life distributions. J. Stat. Comput. Simul. 84, 2619–2635 (2014b).Google Scholar
 Bourguignon, M, Ghosh, I, Cordeiro, GM: General results for the transmuted family of distributions and new models. J. Probab. Statist. Art.ID. 7208425, 21 (2016a).Google Scholar
 Bourguignon, M, Leao, J, Leiva, V, SantosNeto, M: The transmuted BirnbaumSaunders distribution. REVSTAT (2016b). forthcoming.Google Scholar
 Cancho, VG, LouzandaNeto, F, Barriga, GDC: The Poissonexponential lifetime distribution. Comput. Stat. Data Anal. 55, 677–686 (2011a).Google Scholar
 Cancho, VG, Rodrigues, J, deCastro, M: A flexible model for survival data with a cure rate: a Bayesian approach. J. Appl. Statist. 38, 57–70 (2011b).Google Scholar
 Cancho, VG, Louzada, F, Barriga, GDC: The geometric BirnbaumSaunders regression model with cure rate. J. Stat. Plann. Infer. 142, 993–1000 (2012).MathSciNetMATHView ArticleGoogle Scholar
 Cancho, VG, Bandyopadhyay, D, Louzada, F, Yiqi, B: The destructive negarive binomial cure rate model with a latent activation scheme. Stat. Methodol. 13, 48–68 (2013a).Google Scholar
 Cancho, VG, Louzada, F, Ortega, EMM: The power series cure rate model: An application to a cutaneous melanoma data. Commun. Stat. Simul. Comput. 42, 586–602 (2013b).Google Scholar
 Castellares, F, Lemonte, AJ: On the MarshallOlkin extended distributions. Commun. Stat. Theory Methods. 45, 4537–4555 (2016).MathSciNetMATHView ArticleGoogle Scholar
 Chahkandi, M, Gangali, M: On some lifetime distributions with decrasing failure rate. Comput. Stat. Data Anal. 53, 4433–4440 (2009).MATHView ArticleGoogle Scholar
 Chakraborty, S, Bhati, D: Transmuted geometric distribution with applications in modelling and regression analysis of count data. Statist. Oper. Res. Trans. 40, 153–176 (2016).MATHGoogle Scholar
 Chen, MH, Ibrahim, JG, Sinha, D: A new Bayesian model for survival data with a surviving fraction. J. Amer. Statist. Assoc. 94, 909–919 (1999).MathSciNetMATHView ArticleGoogle Scholar
 Chen, Z: A new twoparameter lifetime distribution with bathtub shape or increasing failure rate function. Statist. Probab. Lett. 49, 155–161 (2000).MathSciNetMATHView ArticleGoogle Scholar
 Chung, Y, Kang, Y: The exponentiated Weibullgeometric distribution: Properties and estimations. Commun. Stat. Appl. Methods (Korean). 21, 147–160 (2014).MATHGoogle Scholar
 Cordeiro, GM, Alizadeh, M, Ortega, EMM: The exponentiated halflogistic family of distributions: properties and applications. J. Probab. Statist. Art.ID. 864396, 21 (2014a).Google Scholar
 Cordeiro, GM, deSantana, LH, Ortega, EMM, Pescim, RR: A new family of distributions: LibbyNovick beta. Int. J. Statist. Probab. 3, 63–80 (2014b).Google Scholar
 Cordeiro, GM, Ortega, EMM, Lemonte, A: The exponentialWeibull lifetime distribution. J. Stat. Comput. Simul. 84, 2592–2606 (2014c).Google Scholar
 Cordeiro, GM, Ortega, EMM, Popović, BV, Pescim, RR: The Lomax generator of distributions: Properties, minification process and regression model. Appl. Math. Comput. 247, 465–486 (2014d).Google Scholar
 Cordeiro, GM, Alizadeh, M, Ortega, EMM, Serrano, LHV: The ZografosBalakrishnan odd loglogistic family of distributions: Properties and applications. Hacet. J. Math. Stat. 45 (2015a). doi:http://dx.doi.org/10.15672/HJMS.20159714145. forthcoming.
 Cordeiro, GM, Ortega, EMM, Lemonte, A: The Poisson generalized linear failure rate model. Commun. Stat. Theory Methods. 44, 2037–2058 (2015b).Google Scholar
 Cordeiro, GM, Saboor, A, Khan, MN: The transmuted generalized modified Weibull distribution. Filomat (2015c). forthcoming.Google Scholar
 Cordeiro, GM, Alizadeh, M, Tahir, MH, Mansoor, M, Bourguignon, M, Hamedani, GG: The beta odd loglogistic generalized family of distributions. Hacet. J. Math. Stat. 45, 1175–1202 (2016). forthcoming.Google Scholar
 Cordeiro, GM, Bager, RDSB: Moments for some Kumaraswamy generalized distributions. Commun. Stat. Theory Methods. 44, 2720–2737 (2015).MathSciNetMATHView ArticleGoogle Scholar
 Cordeiro, GM, Bourguignon, M: New results on the RistićBalakrishnan family of distributions. Commun. Stat. Theory Methods. 54, 13–53 (2016).MathSciNetGoogle Scholar
 Cordeiro, GM, deCastro, M: A new family of generalized distributions. J. Stat. Comput. Simul. 81, 883–893 (2011).MathSciNetMATHView ArticleGoogle Scholar
 Cordeiro, GM, Cancho, VG, Ortega, EMM, Barriga, GDC: A model with longterm survivors: negative binomial BirnbaumSaunders. Commun. Stat. Theory Methods. 45, 1370–1387 (2016).MathSciNetMATHView ArticleGoogle Scholar
 Cordeiro, GM, Ortega, EMM, daCunha, DCC: The exponentiated generalized class of distributions. J. Data Sci. 11, 777–803 (2013a).Google Scholar
 Cordeiro, GM, Silva, GO, Ortega, EMM: The beta Weibull geometric distribution. Statistics. 47, 817–834 (2013b).Google Scholar
 Cordeiro, GM, Rodrigues, J, deCastro, M: The exponential COMPoisson distribution. Stat. Papers. 53, 653–664 (2012a).Google Scholar
 Cordeiro, GM, Silva, GO, Ortega, EMM: The beta extended Weibull family. J. Probab. Stat. Sci. 10, 15–40 (2012b).Google Scholar
 Cordeiro, GM, Silva, RB: The complementary extended Weibull power series class of distributions. Ciênc. Nat. 36, 1–13 (2014).MathSciNetGoogle Scholar
 Dagum, C: A new model of personal income distribution: specification and estimation. Econ. Appl. 30, 413–437 (1977).Google Scholar
 Das, KK: On some generalized transmuted distributions. Int. J. Sci. Eng. Res. 6, 1686–1691 (2015).Google Scholar
 daSilva, AL, Rodrigues, J, Silva, G: The beta transmuted Fréchet distribution: Properties and application to survival data (in Portuguese). ForSci. 3, 57–69 (2015a).Google Scholar
 daSilva, RV, GomesSilva, F, Ramos, MWA, Cordeiro, GM: The exponentiated Burr XII Poisson distribution with application to lifetime data. Int. J. Statist. Probab. 4, 112–131 (2015b).Google Scholar
 daSilva, RV, Ramos, MWA, GomesSilva, F, Cordeiro, GM: The exponentiated KumaraswamyG class. J. Egyptian Math. Soc. (2016). forthcoming.Google Scholar
 Delgarm, L, Zadkarami, MR: A new generalization of lifetime distributions. Comput. Statist. 30, 1185–1198 (2015).MathSciNetMATHView ArticleGoogle Scholar
 Dempster, AP, Laird, NM, Rubin, DB: Maximum likelihood from incomplete data via the EM algorithm. J. Roy. Stat. Soc. B 39, 1–38 (1977).Google Scholar
 Ebraheim, AEHN: Exponentiated transmuted Weibull distribution: A generalization of the Weibull distribution. Int. J. Math. Comput. Phys. Quantum Eng. 8, 897–905 (2014).Google Scholar
 Elbatal, I: Transmuted modified inverse Weibull distribution: A generalization of the modified inverse Weibull probability distribution. Int. J. Math. Arch. 4, 117–129 (2013a).Google Scholar
 Elbatal, I: Transmuted generalized inverted exponential distribution. Econ. Qual. Control. 28, 125–133 (2013b).Google Scholar
 Elbatal, I, Aryal, G: On the transmuted additive Weibull distribution. Aust. J. Statist. 42, 117–132 (2013).Google Scholar
 Elbatal, I, Aryal, G: Transmuted Dagum distribution with applications. Chil. J. Statist. 6, 31–45 (2015).MathSciNetGoogle Scholar
 Elbatal, I, Diab, LS, Alim, NAA: Transmuted generalized linear exponential distribution. Int. J. Computer Appl. 83, 29–37 (2013).Google Scholar
 Elbatal, I, Asha, G, Raja, AV: Transmuted exponentiated Fréchet distribution: Properties and applications. J. Statist. Applic. Probab. 3, 379–394 (2014).Google Scholar
 Elbatal, I, Elgarhy, M: Transmuted quasiLindley distribution: A generalization of the quasiLindley distribution. Int. J. Pure Appl. Sci. Technol. 18, 59–70 (2013).Google Scholar
 Elbatal, I, Mansour, MM, Ahsanullah, M: The additive Weibullgeometric distribution: Theory and applications. J. Stat. Theory Applic. 15, 125–141 (2016).View ArticleGoogle Scholar
 Elgarhy, M, Rashed, M, Shawki, AW: Transmuted generalized Lindley distribution. Int. J. Math.– Trends Technol. 29, 145–154 (2016).View ArticleGoogle Scholar
 Eugene, N, Lee, C, Famoye, F: Betanormal distribution and its applications. Commun. Stat. Theory Methods. 31, 497–512 (2002).MathSciNetMATHView ArticleGoogle Scholar
 Fatima, A, Roohi, A: Transmuted exponentiated ParetoI distribution. Pak. J. Statist. 32, 63–80 (2015).MathSciNetGoogle Scholar
 Fioruci, JA, Yiqi, B, Louzada, F, Cancho, VG: The exponential Poisson logarithmic distribution. Commun. Stat. Theory Methods. 45, 2556–2575 (2016).MathSciNetMATHView ArticleGoogle Scholar
 Flores, JD, Borges, P, Cancho, VG, Louzada, F: The complementary exponential power series distribution. Braz. J. Probab. Statist. 27, 565–584 (2013).MathSciNetMATHView ArticleGoogle Scholar
 Gallardo, DI, Bolfarine, H: Two alternative estimation procedures for the negative binomial cure rate model with a latent activation scheme. Statist. Oper. Res. Trans. 40, 31–54 (2016).MathSciNetMATHGoogle Scholar
 Gallardo, DI, Bolfarine, H, PedrosodeLima, AC: An EM algorithm for estimating the destructive weighted Poisson cure rate model. J. Stat. Comput. Simul. 86, 1497–1515 (2016).MathSciNetView ArticleGoogle Scholar
 Gitifar, N, Rezaei, S, Nadarajah, S: Compound distributions motivated by linear failure rate. Statist. Oper. Res. Trans. 40, 177–200 (2016).MATHGoogle Scholar
 Ghorbani, M, Bagheri, SF, Alizadeh, M: A new lifetime distribution: The modified Weibull Poisson distribution. Int. J. Oper. Res. Dec. Sci. Stud. 1, 28–47 (2014).Google Scholar
 Gomes, AE, daSilva, CQ, Cordeiro, GM: The exponentiatedG Poisson model. Commun. Stat. Theory Methods. 44, 4217–4240 (2015).MathSciNetMATHView ArticleGoogle Scholar
 Granzotto, DCT, Louzada, F: The transmuted loglogistic distribution: Modeling, inference, and an application to a polled tabapua race time up to first calving data. Commun. Stat. Theory Methods. 43, 3387–3402 (2015).MathSciNetMATHView ArticleGoogle Scholar
 Gui, W, Zhang, S, Lu, X: The LindleyPoisson distribution in lifetime analysis and its properties. Hacet. J. Math. Stat. 43, 1063–1077 (2014).MathSciNetMATHGoogle Scholar
 Gupta, RC, Gupta, PI, Gupta, RD: Modeling failure time data by Lehmann alternatives. Commun. Stat. Theory Methods. 27, 887–904 (1998).MATHView ArticleGoogle Scholar
 Gupta, RD, Kundu, D: Generalized exponential distribution. Austral. N. Z. J. Stat. 41, 173–188 (1999).MathSciNetMATHView ArticleGoogle Scholar
 Gupta, RC, Wu, Q, Huang, J: Analysis of survival data by an exponentialgeneralized Poisson distribution. J. Stat. Comput. Simul. 84, 2495–2505 (2014).MathSciNetView ArticleGoogle Scholar
 Haq, MA, Butt, NS, Usman, RM, Fattah, AA: Transmuted power function distribution. Ghazi Uni. J. Sci. 29, 177–185 (2016).Google Scholar
 Harandi, SS, Alamatsaz, MA: A complementary generalized linear failure rate geometric distribution. Commun. Stat. Theory Methods. 45, 2204–2227 (2016).MathSciNetMATHView ArticleGoogle Scholar
 Hashimoto, EM, Ortega, EMM, Cordeiro, GM, Cancho, VG: The Poisson BirnbaumSaunders model with longterm survivors. Statistics. 48, 1394–1413 (2014).MathSciNetMATHView ArticleGoogle Scholar
 Hashimoto, EM, Ortega, EMM, Cordeiro, GM, Cancho, VG: A new longterm survival model with intervalcensored data. Sankhyã B 77, 207–239 (2015).Google Scholar
 Hassan, AS, AbdElfattah, AM, Mokhtar, AH: The complementary Burr III Poisson distribution. Austral. J. Basic Appl. Sci. 9, 219–228 (2015).Google Scholar
 Hassan, AS, AbdElfattah, AM, Mokhtar, AH: The complementary exponentiated inverted Weibull power series family of distributions and its application. British J. Math. Comput. Sci. 13, 1–20 (2016).Google Scholar
 Hemmati, F, Khorram, E, Rezakhah, S: A new threeparameter ageing distribution. J. Stat. Plann. Infer. 141, 2266–2275 (2011).MathSciNetMATHView ArticleGoogle Scholar
 Hussian, MA: Estimation of P(Y<X) for the class of KumaraswamyG distributions. Austral. J. Basic Appl. Sci. 7, 158–169 (2013).Google Scholar
 Hussian, MA: Transmuted exponentiated gamma distribution: A generalization of exponentiated gamma probability distribution. Appl. Math. Sci. 27, 1297–1310 (2014).MathSciNetView ArticleGoogle Scholar
 Iriarte, YA, Astorga, JM: Transmuted Maxwell probability distribution (in Portuguese). Rev. Integr. 32, 211–221 (2014).MathSciNetMATHGoogle Scholar
 Iriarte, YA, Astorga, JM: A version of transmuted generalized Rayleigh distribution (in Portuguese). Rev. Integr. 33, 83–95 (2015).MathSciNetMATHGoogle Scholar
 Jafari, AA, Tahmasebi, S: Gompertzpower series distributions. Commun. Stat. Theory Methods. 45, 3761–3781 (2016).MathSciNetMATHView ArticleGoogle Scholar
 Jiménez, JA, Arunachalam, V, Serna, GM: A generalization of Tukey’s gh family of distributions. J. Stat. Theory Applic. 14, 28–44 (2015).View ArticleGoogle Scholar
 Kannan, N, Kundu, D, Nair, P, Tripathi, RC: The generalized exponential cure rate model with covariates. J. Appl. Statist. 37, 1625–1636 (2010).MathSciNetView ArticleGoogle Scholar
 Khan, MS, King, R: Transmuted modified Weibull distribution: A generalization of the modified Weibull probability distribution. Eur. J. Pure Appl. Math. 6, 66–88 (2013).MathSciNetGoogle Scholar
 Khan, MS, King, R: A new class of transmuted inverse Weibull distribution for reliability analysis. Amer. J. Math. Manag. Sci. 33, 261–286 (2014a).Google Scholar
 Khan, MS, King, R: Transmuted generalized inverse Weibull distribution. J. Appl. Stat. Sci. 20, 213–230 (2014b).Google Scholar
 Khan, MS, King, R: Transmuted modified inverse Rayleigh distribution. Aust. J. Statist. 44, 17–29 (2015).View ArticleGoogle Scholar
 Khan, MS, King, R, Hudson, IL: Characterizations of the transmuted inverse Weibull distribution. ANZIAM J. 55, C197—C217 (2014).Google Scholar
 Khan, MS, King, R, Hudson, IL: Transmuted generalized exponential distribution: A generalization of the exponential distribution with applications to survival data. Commun. Stat. Theory Methods (2015a). forthcoming.Google Scholar
 Khan, MS, King, R, Hudson, IL: A new three parameter transmuted Chen lifetime distribution with application. J. Appl. Stat. Sci. 21, 239–259 (2015b).Google Scholar
 Khan, MS, King, R, Hudson, IL: Transmuted Kumaraswamy distribution. Statist. Transition. 17, 1–28 (2016a).Google Scholar
 Khan, MS, King, R, Hudson, IL: Transmuted Weibull distribution: Properties and estimation. Commun. Stat. Theory Methods (2016b). doi:http://dx.doi.org/10.1080/03610926.2015.1100744. forthcoming.
 Khan, MS, King, R, Hudson, IL: Transmuted Gompertz distribution: Properties and estimation. Pak. J. Statist. 32, 161–182 (2016c).Google Scholar
 Kuş, C: A new lifetime distribution. Comput. Stat. Data Anal. 51, 4497–4509 (2007).MathSciNetMATHView ArticleGoogle Scholar
 Lai, CD, Xie, M, Murthy, DNP: A modified Weibull distribution. IEEE Trans. Reliab. 52, 33–37 (2003).View ArticleGoogle Scholar
 Leahu, A, Munteanu, BG, Cataranciuc, S: On the lifetime as the maximum or minimum of the sample with power series distributed size. ROMAI J. 9, 119–128 (2013).MathSciNetMATHGoogle Scholar
 Leahu, A, Munteanu, BG, Cataranciuc, S: MaxErlang and MinErlang power series distributions as two new families of lifetime distribution. Bull. Acad. Ştiinţe. 2, 60–73 (2014).MathSciNetMATHGoogle Scholar
 Lemonte, AJ, Cordeiro, GM, Ortega, EMM: On the Additive Weibull Distribution. Commun. Stat. Theory Methods. 43, 2066–2080 (2014).MathSciNetMATHView ArticleGoogle Scholar
 Louzada, F, Bereta, EMP, Franco, MAP: On the distribution of the minimum or maximum of a random number of i.i.d. lifetime random variable. Appl. Math. 3, 350–353 (2012a).Google Scholar
 Louzada, F, Borges, P, Cancho, V: The exponential negativebinomial distribution: A continuous bridge between under and over dispersion on a lifetime modelling structure. J. Statist. Adv. Theory Applic. 7, 67–83 (2012b).Google Scholar
 Louzada, F, Cancho, VG, Barriga, GDC: The Poissonexponential regression model under different latent activation schemes. Comput. Appl. Math. 31, 617–632 (2012c).Google Scholar
 Louzada, F, Cancho, VG, Roman, M, Leite, JG: A new longterm lifetime distribution induced by a latent complementary risk framework. J. Appl. Statist. 39, 2209–2222 (2012d).Google Scholar
 Louzada, F, Cancho, VG, Yiqi, B: The logWeibullnegativebinomial regression model under latent failure causes and presence of randomized schemes. Statistics. 49, 930–949 (2015).MathSciNetMATHView ArticleGoogle Scholar
 Louzada, F, Granzotto, DCT: The transmuted loglogistic regression model: a new model for time up to first calving of cows. Stat. Papers (2015). doi:http://dx.doi.org/10.1007/s0036201506715. forthcoming.
 Louzada, F, Marchi, V, Carpenter, J: The complementary exponentiated exponential geometric lifetime distribution. J Probab Statist. Art.ID. 502159, 12 (2014a).Google Scholar
 Louzada, F, Marchi, V, Romana, M: The exponentiated exponential geometric distribution: a distribution with decreasing, increasing and unimodel failure rate. Statistics. 48, 167–181 (2014).MathSciNetMATHView ArticleGoogle Scholar
 Louzada, F, Roman, M, Cancho, VG: The complementary exponential geometric distribution: Model, properties, and a comparison with its counterpart. Comput. Stat. Data Anal. 55, 2516–2524 (2011).MathSciNetView ArticleGoogle Scholar
 Lu, W, Shi, D: A new compounding life distribution: the Weibull Poisson distribution. J. Appl. Statist. 39, 21–38 (2012).MathSciNetView ArticleGoogle Scholar
 Lucena, SEF, Silva, AHA, Cordeiro, GM: The transmuted generalized gamma distribution: Properties and application. J. Data Sci. 13, 409–420 (2015).Google Scholar
 Luguterah, A, Nasiru, S: Transmuted exponential Pareto distribution. Far East. J. Theor. Statist. 50, 31–49 (2015).MathSciNetMATHView ArticleGoogle Scholar
 Mahmoud, MR, Mandouh, RM: On the transmuted Fréchet distribution. J. Appl. Sci. Res. 9, 5553–5561 (2013).Google Scholar
 Mahmoudi, E, Jafari, AA: Generalized exponential–power series distributions. Comput. Stat. Data Anal. 56, 4047–4066 (2012).MathSciNetMATHView ArticleGoogle Scholar
 Mahmoudi, E, Jafari, AA: The compound class of linear failure ratepower series distributions: Model, properties and applications. Commun. Stat. Simul. Comput. (2015). doi:http://dx.doi.org/10.1080/03610918.2015.1005232. forthcoming.
 Mahmoudi, E, Mahmoodian, H: Normal power series class of distributions: Model, properties and applications (2015). arXiv:1510.07180v1 [stat.CO].Google Scholar
 Mahmoudi, E, Sepahdar, A: Exponentiated WeibullPoisson distribution: Model, properties and applications. Math. Comput. Simul. 92, 76–97 (2013).MathSciNetView ArticleGoogle Scholar
 Mahmoudi, E, Sepahdar, A, Lemonte, AJ: Exponentiated Weibulllogarithmic distribution: Model, properties and applications (2014). arXiv:1402.2564v1 [stat.ME].Google Scholar
 Mahmoudi, E, Shiran, M: Exponentiated Weibullgeometric distribution and its applications (2012a). arXiv:1206.4008v1 [stat.CO].Google Scholar
 Mahmoudi, E, Shiran, M: Exponentiated Weibullpower series distributions and its applications (2012b). arXiv:1212.5613v1 [stat.CO].Google Scholar
 Mansour, MM, Elrazik, EMB, Hamed, MS, Mohamed, SM: A new transmuted additive Weibull distribution: Based on a new method for adding a parameter to a family of distribution. Int. J. Appl. Math. Sci. 8, 31–51 (2015a).Google Scholar
 Mansour, MM, Hamed, MS, Mohamed, SM: A new Kumaraswamy transmuted modified Weibull distribution with application. J. Stat. Adv. Theory Applic. 13, 101–133 (2015b).Google Scholar
 Mansour, MM, Mohamed, SM: A new generalized of transmuted Lindley distribution. Appl. Math. Sci. 9, 2729–2748 (2015).View ArticleGoogle Scholar
 Marshall, AW, Olkin, I: A new method for adding a parameter to a family of distributions with application to the exponential and Weibull families. Biometrika. 84, 641–652 (1997).MathSciNetMATHView ArticleGoogle Scholar
 Martinez, EZ, Achcar, JA, Jácome, AAA, Santos, JS: Mixture and nonmixture cure fraction models based on the generalized modified Weibull distribution with an application to gastric cancer data. Comput. Methods Prog. Biomed. 112, 343–355 (2013).Google Scholar
 Mendoza, NVR, Ortega, EMM, Cordeiro, GM: The exponentiatedloglogistic geometric distribution: Dual activation. Commun. Stat. Theory Methods. 45, 3838–3859 (2016).MathSciNetMATHView ArticleGoogle Scholar
 Merovci, F: Transmuted Rayleigh distribution. Aust. J. Statist. 42, 21–31 (2013a).Google Scholar
 Merovci, F: Transmuted exponentiated exponential distribution. Math. Sci. Applic. ENotes. 1, 112–122 (2013b).Google Scholar
 Merovci, F: Transmuted Lindley distribution. Int. J. Open Problems Comput. Math. 6, 63–72 (2013c).Google Scholar
 Merovci, F: Transmuted generalized Rayleigh distribution. J. Stat. Applic. Probab. 3, 9–20 (2014).View ArticleGoogle Scholar
 Merovci, F, Elbatal, I, Ahmed, A: The transmuted generalized inverse Weibull distribution. Aust. J. Statist. 43, 119–131 (2014).View ArticleGoogle Scholar
 Merovci, F, Alizadeh, M, Hamedani, GG: Another generalized transmuted family of distributions: Properties and applications. Aust. J. Statist. 45, 71–93 (2016).View ArticleGoogle Scholar
 Merovci, F, Elbatal, I: Transmuted Lindleygeometric distribution and its aplications. J. Stat. Applic. Probab. 3, 77–91 (2014a).Google Scholar
 Merovci, F, Elbatal, I: Transmuted Weibullgeometric distribution and its aplications. Sci. Magna. 10, 68–82 (2014b).Google Scholar
 Merovci, F, Puka, L: Transmuted Pareto distribution. ProbStat Forum. 7, 1–11 (2014).MathSciNetMATHGoogle Scholar
 Morais, AL, BarretoSouza, W: A compound class of Weibull and power series distributions. Comput. Stat. Data Anal. 55, 1410–1425 (2011).MathSciNetMATHView ArticleGoogle Scholar
 Munteanu, BG: The maxWeibull power series distribution. An. Univ. Oradea Fasc. Mat. 21, 133–139 (2014).MathSciNetMATHGoogle Scholar
 Nadarajah, S, Cancho, VG, Ortega, EMM: The geometric exponential Poisson distribution. Stat. Methods Applic. 22, 355–380 (2013a).Google Scholar
 Nadarajah, S, Jayakumar, K, Ristić, MM: A new family of lifetime models. J. Stat. Comput. Simul. 83, 1389–1404 (2013b).Google Scholar
 Nadarajah, S, Cordeiro, GM, Ortega, EMM: General results for the KumaraswamyG distribution. J. Stat. Comput. Simul. 82, 951–979 (2012).MathSciNetMATHView ArticleGoogle Scholar
 Nadarajah, S, Cordeiro, GM, Ortega, EMM: The exponentiatedGgeometric family of distributions. J. Stat. Comput. Simul. 85, 1634–1650 (2015a).Google Scholar
 Nadarajah, S, Cordeiro, GM, Ortega, EMM: The ZografosBalakrishnanG family of distributions: Mathematical properties and application. Commun. Stat. Theory Methods. 44, 186–215 (2015b).Google Scholar
 Nadarajah, S, Shahsanaei, F, Rezaei, S: A new fourparameter lifetime distribution. J. Stat. Comput. Simul. 84, 248–263 (2014a).Google Scholar
 Nadarajah, S, Teimouri, M, Shih, SH: Modified beta distributions. Sankhyã B 76, 19–48 (2014b).Google Scholar
 Nascimento, ADC, Bourguignon, M, Zea, LM, SantosNeto, M, Silva, RB, Cordeiro, GM: The gamma extended Weibull family of distributions. J. Stat. Theory Applic. 13, 1–16 (2014).MathSciNetMATHView ArticleGoogle Scholar
 Nassar, M, Nada, N: A new generalization of the exponentialgeometric distribution. J. Statist. Adv. Theory Applic. 7, 25–48 (2012).MATHGoogle Scholar
 Nofal, Z, Afify, A, Yousof, H, Cordeiro, GM: The generalized transmutedG family of distributions. Commun. Stat. Theory Methods (2016). doi:http://dx.doi.org/10.1080/03610926.2015.1078478. forthcoming.
 Oluyede, BO, Foya, S, WarahenaLiyanage, G, Huang, S: The loglogistic Weibull distribution with applications to lifetime data. Aust. J. Statist. 45, 43–69 (2016a).Google Scholar
 Oluyede, BO, Motsewabagale, G, Huang, S, WarahenaLiyanage, G, Rararai, M: The DagumPoisson distribution: model, properties and application. Electron. J. Appl. Stat. Anal. 9, 169–197 (2016b).Google Scholar
 Oluyede, BO, WarahenaLiyanage, G, Rararai, M: A new compund class of loglogistic WeibullPoisson distribution: model, properties and application. J. Stat. Comput. Simul. 86, 1363–1391 (2016c).Google Scholar
 Ortega, EMM, Cordeiro, GM, Kattan, MW: The negative binomialbeta Weibull regresion model to predict the cure rate of prostrate cancer. J. Appl. Statist. 39, 1191–1210 (2012).MathSciNetView ArticleGoogle Scholar
 Ortega, EMM, Barriga, GDC, Hashimoto, EM, Cancho, VG, Cordeiro, GM: A new class of survival regression models with cure fraction. J. Data Sci. 12, 107–136 (2014).MathSciNetGoogle Scholar
 Ortega, EMM, Cordeiro, GM, Campelo, AK, Kattan, MW, Cancho, VG: A power series beta Weibull regression model for predicing breast carcinoma. Statist. Med. 34, 1366–1388 (2015).MathSciNetView ArticleGoogle Scholar
 Owoloko, EA, Oguntunde, PE, Adejumo, AO: Performance rating of the transmuted exponential distribution: an analytical approach. SpringerPlus. 4, 8–18 (2015).View ArticleGoogle Scholar
 Pal, M, Tiensuwan, M: The beta transmuted Weibull distribution. Aust. J. Statist. 43, 133–149 (2014).View ArticleGoogle Scholar
 Pappas, V, Adamidis, K, Loukas, S: A threeparameter lifetime distribution. Adv. Applic. Statist. 20, 159–167 (2011).MathSciNetMATHGoogle Scholar
 Pappas, V, Adamidis, K, Loukas, S: A generalization of the exponentiallogarithmic distribution. J. Stat. Theory Prac. 9, 122–133 (2015).MathSciNetView ArticleGoogle Scholar
 Pararai, M, Oluyede, BO, WarahenaLiyanage, G: Kumaraswamy LindleyPoisson distribution: Theory and applications. Asian. J. Math. Appl. Art. ID. ama0261, 30 (2015a).Google Scholar
 Pararai, M, WarahenaLiyanage, G, Oluyede, BO: An extended Lindley Poisson distribution with applications. J. Math. Stat. Sci. 1, 167–198 (2015b).Google Scholar
 Pararai, M, WarahenaLiyanage, G, Oluyede, BO: Exponentiated power Lindley Poisson distribution: properties and applications. Commun. Stat. Theory Methods (2016). doi:http://dx.doi.org/10.1080/03610926.2015.1076473. forthcoming.
 Pescim, RR, Cordeiro, GM, Demétrio, CGB, Ortega, EMM, Nadarajah, S: The new class of Kummer beta generalized distributions. Statist. Oper. Res. Trans. 36, 153–180 (2012).MathSciNetMATHGoogle Scholar
 Pinho, LGB, Cordeiro, GM, Nobre, JS: On the HarrisG class of distributions: general results and application. Braz. J. Probab. Statist. 29, 813–832 (2015).MathSciNetMATHView ArticleGoogle Scholar
 Preda, V, Panaitescu, E, Ciumara, R: The modified exponentialPoisson distribution. Proc. Rom. Acad. A12, 22–29 (2011).MathSciNetMATHGoogle Scholar
 Percontini, A, Blas, B, Cordeiro, GM: The beta Weibull Poisson distribution. Chil. J. Statist. 4, 3–26 (2013a).Google Scholar
 Percontini, A, Cordeiro, GM, Bourguignon, M: The Gnegative binomail family: General properties and applications. Adv. Applic. Statist. 35, 127–160 (2013b).Google Scholar
 Percontini, A, GomesSilva, FS, Ramos, MWA, Venancio, R, Cordeiro, GM: The gamma Weibull Poisson distribution applied to survival data. Trends Appl. Comput. Math. (TEMA). 15, 165–176 (2014).MathSciNetGoogle Scholar
 Popović, BV, Ristić, MM, Cordeiro, GM: A two parameter distribution obtained by compounding the generalized exponential and exponential distributions. Mediterr. J. Math (2015). doi:http://dx.doi.org/10.1007/s0000901506655. forthcoming.
 Ramos, MWA, Marinho, PRD, Cordeiro, GM, daSilva, RV, Hamedani, GG: The KumaraswamyG Poisson family of distributions. J. Stat. Theory Applic. 14, 222–239 (2015).MathSciNetView ArticleGoogle Scholar
 Ramos, MWA, Marinho, PRD, daSilva, RV, Cordeiro, GM: The exponentiated Lomax Poisson distribution with an application to lifetime data. Adv. Applic. Statist. 34, 107–135 (2013).MathSciNetMATHGoogle Scholar
 Rezaei, S, Tahmasbi, R: A new lifetime distribution with increasing failure rate: Exponential truncated Poisson. J. Basic Appl. Sci. Res. 2, 1749–1762 (2012).Google Scholar
 Ristić, MM, Balakrishnan, N: The gammaexponentiated exponential distribution. J. Stat. Comput. Simul. 82, 1191–1206 (2012).MathSciNetMATHView ArticleGoogle Scholar
 Ristić, MM, Kundu, D: Generalized exponential geometric extreme distribution. J. Stat. Theory Prac. 10, 179–201 (2016).MathSciNetView ArticleGoogle Scholar
 Ristić, MM, Nadarajah, S: A new lifetime distribution. J. Stat. Comput. Simul. 84, 135–150 (2014).MathSciNetView ArticleGoogle Scholar
 Ristić, MM, Popvić, BV, Nadarajah, S: Libby and Novick’s generalized beta exponential distribution. J. Stat. Comput. Simul. 85, 740–761 (2015).MathSciNetView ArticleGoogle Scholar
 Rodrigues, J, Cancho, VG, deCastro, M, LouzadaNeto, F: On the unification of longterm survival models. Statist. Probab. Lett. 79, 753–759 (2009a).Google Scholar
 Rodrigues, J, deCastro, M, Cancho, VG, Balakrishnan, N: COMPoisson cure rate survival models and an application to a cutaneous melanoma data. J. Stat. Plann. Infer. 139, 3605–3611 (2009b).Google Scholar
 Roman, M, Louzada, F, Cancho, VG, Leite, JG: A new longterm survival distribution for cancer data. J. Data Sci. 10, 241–258 (2012).MathSciNetGoogle Scholar
 Saboor, A, Kamal, M, Ahmad, M: The transmuted exponentiatedWeibull distribution with applications. Pak. J. Statist. 31, 229–250 (2015).MathSciNetGoogle Scholar
 Saboor, A, Elbatal, I, Cordeiro, GM: The transmuted exponentiated Weibull geometric distribution: Theory and applications. Hacet. J. Math. Stat. 45, 973–987 (2016).Google Scholar
 Sarhan, AM, Kundu, D: Generalized linear failure rate distribution. Commun. Stat. Theory Methods. 38, 642–660 (2009).MathSciNetMATHView ArticleGoogle Scholar
 Sarhan, AM, Zaindin, M: Modified Weibull distribution. Appl. Sci. 11, 123–136 (2009).MathSciNetMATHGoogle Scholar
 Shahsanaei, F, Rezaei, S, Pak, A: A new twoparameter lifetime distribution with increasing failure rate. Econ. Qual. Control. 27, 1–17 (2012).MATHView ArticleGoogle Scholar
 Shafiei, S, Darijani, S, Saboori, H: Inverse Weibull power series distributions: properties and applications. J. Stat. Comput. Simul. 86, 1069–1094 (2016).MathSciNetView ArticleGoogle Scholar
 Sharma, VK, Singh, SK, Singh, U: A new upsidedown bathtub shaped hazard rate model for survival data analysis. Appl. Math. Comput. 239, 242–253 (2014).MathSciNetMATHGoogle Scholar
 Shaw, WT, Buckley, IR: The alchemy of probability distributions: Beyond GramCharlier expansions, and a skewkurtoticnormal distribution from a rank transmutation map (2009). arXiv:0901.0434 [qfin.ST].Google Scholar
 Shahzad, MN, Asghar, Z: Transmuted Dagum distribution: A more flexible and broad shaped hazard function model. Hacet. J. Math. Stat. 45, 227–224 (2016).MATHGoogle Scholar
 Silva, RB, BarretoSouza, W, Cordeiro, GM: A new distribution with decreasing, increasing and upsidedown bathtub failure rate. Comput. Stat. Data Anal. 54, 935–944 (2010).MathSciNetMATHView ArticleGoogle Scholar
 Silva, RB, Bourguignon, M, Dias, CRB, Cordeiro, GM: The compound class of extended Weibull power series distributions. Comput. Stat. Data Anal. 58, 352–367 (2013).MathSciNetView ArticleGoogle Scholar
 Silva, RB, Bourguignon, M, Cordeiro, GM: A new compounding family of distributions: The generalized gamma power series distributions. J. Comput. Appl. Math. 303, 119–139 (2016).MathSciNetMATHView ArticleGoogle Scholar
 Silva, RB, Cordeiro, GM: The Burr XII power series distributions: A new compounding family. Braz. J. Probab. Statist. 29, 565–589 (2015).MathSciNetMATHView ArticleGoogle Scholar
 Tahir, MH, Cordeiro, GM, Alizadeh, M, Mansoor, M, Zubair, M, Hamedani, GG: The odd generalized exponential family of distributions with applications. J. Stat. Dist. Applic. 2, Art, 1 (2015).Google Scholar
 Tahir, MH, Cordeiro, GM, Alzaatreh, A, Mansoor, M, Zubair, M: The LogisticX family of distributions and its applications. Commun. Stat. Theory Methods. 45, 732–7349 (2016). forthcoming.MathSciNetGoogle Scholar
 Tahir, MH, Nadarajah, S: Parameter induction in continuous univariate distribution: Wellestablished G families. Ann. Braz. Acad. Sci. 87, 539–568 (2015).MathSciNetView ArticleGoogle Scholar
 Tahir, MH, Zubair, M, Cordeiro, GM, Alzaatreh, A, Mansoor, M: The PoissonX family of distributions. J. Stat. Comput. Simul. 86, 2901–2921 (2016a).Google Scholar
 Tahir, MH, Zubair, M, Mansoor, M, Cordeiro, GM, Alizadeh, M, Hamedani, GG: A new WeibullG family of distributions. Hacet. J. Math. Stat. 45, 629–647 (2016b).Google Scholar
 Tahmasbi, R, Rezaei, S: A twoparameter lifetime distribution with decreasing failure rate. Comput. Stat. Data Anal. 52, 3889–3901 (2008).MathSciNetMATHView ArticleGoogle Scholar
 Tahmasebi, S, Jafari, AA: Exponentiated extended Weibullpower series class of distributions. Ciênc. Nat. 37, 183–193 (2015a).Google Scholar
 Tahmasebi, S, Jafari, AA: Generalized Gompertzpower series distributions. Hacet. J. Math. Stat. (2015b). doi:http://dx.doi.org/10.15672/HJMS.20158312681. forthcoming.
 Tian, Y, Tian, M, Zhu, Q: Transmuted linear exponential distribution: A new generalization of the linear exponential distribution. Commun. Stat. Simul. Comput. 43, 2661–2671 (2014).MathSciNetMATHView ArticleGoogle Scholar
 Tojeiro, C, Louzada, F, Roman, M, Borges, P: The complementary Weibull geometric distribution. J. Stat. Comput. Simul. 84, 1345–1362 (2014).MathSciNetView ArticleGoogle Scholar
 Torabi, H, Montazari, NH: The gammauniform distribution and its application. Kybernetika. 48, 16–30 (2012).MathSciNetGoogle Scholar
 Torabi, H, Montazari, NH: The logisticuniform distribution and its application. Commun. Stat. Simul. Comput. 43, 2551–2569 (2014).MATHView ArticleGoogle Scholar
 Triantafyllou, IS, Koutras, MV: Failure rate and aging properties of generalized beta and gammagenerated distributions. Commun. Stat. Theory Methods. 43, 4046–4061 (2014).MathSciNetMATHView ArticleGoogle Scholar
 Vardhan, RV, Balaswamy, S: Transmuted new modified Weibull distribution. Math. Sci. Applic. ENotes. 4, 125–135 (2016).Google Scholar
 Wang, M: A new threeparameter lifetime distribution and associated inference (2013). arXiv:1308.4128v1 [stat.ME].Google Scholar
 Wang, M, Elbatal, I: The modified Weibull geometric distribution. Metron. 73, 303–315 (2015).MathSciNetMATHView ArticleGoogle Scholar
 WarahenaLiyanage, G, Pararai, M: The Lindley power series class of distributions: Model, properties and applications. J. Comput. Model. 5, 35–80 (2015a).Google Scholar
 WarahenaLiyanage, G, Pararai, M: A generalized power Lindley Poisson distribution with applications. Asian J. Math. Appl. Art.ID. ama0169, 23 (2015b).Google Scholar
 Xie, M, Lai, CD: Reliability analysis using an additive Weibull model with bathtub shaped failure rate function. Reliab. Eng. Syst. Safe. 52, 87–93 (1995).View ArticleGoogle Scholar
 Yakovlev, AY, Tsodikov, AD: Stochastic Models of Tumor Latency and Their Biostatistical Applications. World Scientific, Singapore (1996).MATHView ArticleGoogle Scholar
 Yamachi, CY, Romana, M, Louzada, F, Franco, MAP, Cancho, VG: The exponentiated complementary exponential geometric distribution. J. Mod. Math. Front. 2, 78–83 (2013).Google Scholar
 Yiqi, B, Russo, CM, Cancho, VG, Louzada, F: Influence diagnostics for the Weibullnegativebinomial regression model with cure rate under latent failure causes. J. Appl. Statist. 43, 1027–1060 (2016).MathSciNetView ArticleGoogle Scholar
 Yousof, HM, Afify, AZ, Alizadeh, M, Butt, NS, Hamedani, GG, Ali, MM: The transmuted exponentiated generalizedG family of distributions. Pak. J. Statist. Oper. Res. 11, 441–464 (2015).MathSciNetView ArticleGoogle Scholar
 Yousof, HM, Afify, AZ, Ebraheim, AEHN, Hamedani, GG, Butt, NS: On sixparameter Fréchet distribution: Properties and applications. Pak. J. Statist. Oper. Res. 12, 281–299 (2016).MathSciNetView ArticleGoogle Scholar
 Zakerzadeh, H, Mahmoudi, E: A new two parameter lifetime distribution: Model and properties (2013). arXiv:1204.4248v1 [stat.CO].Google Scholar
 Zografos, K, Balakrishnan, N: On families of beta and generalized gammagenerated distributions and associated inference. Stat. Methodol. 6, 344–362 (2009).MathSciNetMATHView ArticleGoogle Scholar