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
Compound Gclasses generalized classes Ggeometric class GPoisson class power series classAMS Subject Classification
Primary 60E05 secondary 62N05 62F10Introduction
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
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