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Compounding of distributions: a survey and new generalized classes
Journal of Statistical Distributions and Applications volumeÂ 3, ArticleÂ number:Â 13 (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.
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 (RistiÄ‡ 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; RistiÄ‡ 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 Tukeyâ€™s g and hG (JimÃ©nez 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).
For any baseline distribution cdf G(x), the cdf of the transmutedG class (for Î·â‰¤1) is given by
where Î· is the transmuted (or shape) parameter. For Î·=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).
More than fifty transmuted distributions have been reported in the literature, which are complied in chronological order in Table 1; see also Section 8.4.
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},â€¦,Y _{ N }) (the minimum of an unknown number of any continuous random variables) or Z= max(Y _{1},â€¦,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 Î± 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,â€¦). We consider that the failure times of the components Z _{ i,1},â€¦,Z _{ i,Î± } 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 Ï„, say T[G(x;Ï„),Î±]=G(x;Ï„)^{Î±} (for x>0). In the following construction, although Î± is a positive integer called power or resilience parameter, we can consider that Î±>0.
If we define Y= min{Y _{1},â€¦,Y _{ N }}, then the conditional cdf of Y given N is
If we define Z= max{Y _{1},â€¦,Y _{ N }}, then the conditional (or complementary) cdf of Z given N is
The unconditional cdf of Y follows from Eq. (2) as
The unconditional cdf of Z follows from Eq. (3) as
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},â€¦,Y _{ N }} and Z= max{Y _{1},â€¦,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
If T[G(y;Ï„);Î±]=G(y;Ï„)^{Î±} and N be a ZTG r.v. with pmf \(\mathbb {P}(N=n)=q\,p^{n1},\, n=1,2,\dots,\,p\in (0,1)\), where p is the success probability, the cdf of the exponentiated Ggeometric (EGG) class can follow as
The EGG class has recently been introduced by Nadarajah et al. (2015a), and can also be called the generalized Ggeometric class. For Î±=1, the EGG class reduces to the Ggeometric (GG) class proposed by Alkarni (2012). They investigated some of its general properties.
Remark 1
Swapping p with q in the ZTG pmf with p+q=1, the cdf of an alternative EGG (denoted by EGGA) class is given by
For Î±=1, the EGGA class reduces to the alternative Ggeometric (GGA) class defined by Castellares and Lemonte ( 2016 ).
3.2 Exponentiated KumaraswamyGgeometric class (new)
For any baseline cdf G(x) and pdf g(x), Cordeiro and deCastro (2011) defined the cdf of the KumaraswamyG class by K _{ a,b }(x;Ï„)=1âˆ’[1âˆ’G(x)^{a}]^{b}, where a>0 and b>0 are both shape parameters. Let T[G(y;Ï„);Î±]=K _{ a,b }(y;Ï„)^{Î±} and N be a ZTG r.v. with pmf \(\mathbb {P}(N=n)=q\, p^{n1},\, n=1,2,\dots \), the cdf of the exponentiated KumaraswamyGgeometric (EKGG) class can follow as
For Î±=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.
If Î±=1 and T[G(y;Ï„)]=M _{ a,b,c }(y;Ï„), and N be a ZTG r.v. with pmf \(\mathbb {P}(N=n)=q\, p^{n1},\,n=1,2,\dots \), the cdf of the McDonaldG geometric (MGG) class is given by
3.4 BetaGgeometric class (new)
For any baseline cdf G(x), Eugene et al. (2002) defined the cdf of the betaG class by
If Î±=1 and T[G(y;Ï„]=B _{ a,b }(y;Ï„), and N be a ZTG r.v. with pmf \(\mathbb {P}(N=n)=q\, p^{n1},\, n=1,2,\dots \), the cdf of the betaG geometric (BGG) class is given by
3.5 Exponentiated GPoisson class
If T[G(y;Ï„);Î±]=G(y;Ï„)^{Î±} and N be a ZTP r.v. with parameter Î» defined by the pmf \(\mathbb {P}(N=n)=\lambda ^{n}/[n!\,(\mathrm {e}^{\lambda }1)],\,n= 1,2,\ldots \), the cdf of the exponentiated GPoisson (EGP) class can follow as
The EGP class has been studied by Gomes et al. (2015). For Î±=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)
If T[G(y;Ï„);Î±]=G(y;Ï„)^{Î±} and N be a ZTP r.v. with parameter Î», the cdf of the exponentiated KumaraswamyG Poisson (EKGP) class is given by
For Î±=1, the EKGP class reduces to the KumaraswamyG Poisson (KGP) family studied by Ramos et al. (2015).
3.7 McDonaldGPoisson class (new)
Let Î±=1 and T[G(y;Ï„)]=M _{ a,b,c }(y;Ï„) and N be a ZTP r.v. with parameter Î», the cdf of the McDonaldG Poisson (MGP) class can be expressed as
3.8 BetaGPoisson class (new)
If Î±=1 and T[G(y);Ï„]=B _{ a,b }(y;Ï„) and N be a ZTP r.v. with parameter Î», the cdf of the betaG Poisson (BGP) class reduces to
3.9 Exponentiated Glogarithmic class (new)
If T[G(y;Ï„);Î±]=G(y;Ï„)^{Î±} and N be a Ln r.v. with pmf \(\mathbb {P}(N=n)=(1\phi)^{n}/[n\,\ln \,\phi ],\,\phi \in (0,1),\, n=1, 2,\dots \), the cdf of the exponentiated Glogarithmic (EGLn) class is given by
by noting that \(\sum _{n=1}^{\infty }\,Q^{n}/n=\,\ln (1Q)\).
For Î±=1, the EGLn class becomes the Glogarithmic (GLn) family introduced by Alkarni (2012).
3.10 Exponentiated KumaraswamyGlogarithmic class (new)
If T[G(y;Ï„);Î±]=G(y;Ï„)^{Î±} and N be a Ln r.v. with pmf \(\mathbb {P}(N=n)=(1\phi)^{n}/[n\,\ln \,\phi ],\, \phi \in (0,1),\quad n=1, 2,\dots \), the cdf of the exponentiated KumaraswamyGlogarithmic (EKGLn) is given by
For Î±=1, the EKGLn cdf is identical to the cdf of the new KumaraswamyGlogarithmic (KGLn) class.
3.11 McDonaldGlogarithmic class (new)
If T[G(y;Ï„)]=M _{ a,b,c }(y;Ï„) and N be a Ln r.v. with pmf \(\mathbb {P}(N=n)=(1\phi)^{n}/[n\,\ln \,\phi ],\, \phi \in (0,1),\quad n=1, 2,\dots \), the cdf of the McDonaldGlogarithmic (MGLn) class is defined by
3.12 BetaGlogarithmic class (new)
If Î±=1 and T[G(y;Ï„)]=B _{ a,b }(y;Ï„) and N be a Ln r.v. with pmf \(\mathbb {P}(N=n)=(1\phi)^{n}/[n\,\ln \,\phi ],\, \phi \in (0,1),\, n=1, 2,\dots \), the cdf of the betaGlogarithmic (BGLn) class is given by
3.13 Exponentiated Gbinomial class (new)
If T[G(y;Ï„);Î±]=G(y;Ï„)^{Î±} and N be a ZTBi r.v. with pmf given by
the cdf of the exponentiated Gbinomial (EGBi) class reduces to
Bakouch et al. (2012b) studied a special case of the EGBi class socalled the exponentiatedexponential binomial (EEBi) model. For Î±=1, the EGBi class becomes the Gbinomial (GBi) class pioneered by Alkarni (2013).
3.14 Exponentiated GNB class (new)
If T[G(y;Ï„);Î±]=G(y;Ï„)^{Î±} and N be a ZTNB r.v. with parameter Î²âˆˆ(0,1) as the probability of success and pmf given by
the cdf of the exponentiated GNB (EGNB) class can follow as
Using the binomial expansion \((1w)^{s}1=\sum _{n=1}^{\infty } \binom {s+n1}{n}\, w^{n}\), we obtain
For Î±=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 Î² by 1âˆ’Î² and the dispersion parameter s by Î¸ 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 Î±=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 Î² 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 Î±=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)
If T[G(z;Ï„);Î±]=G(z;Ï„)^{Î±} and N be a ZTG r.v. with geometric pmf \(\mathbb {P}(N=n)=q\, p^{n1},\, n=1, 2,\dots \phantom {\dot {i}\!}\), the cdf of the complementary exponentiated Ggeometric (CEGG) class reduces to
This equation is also called the complementary generalized Ggeometric family. For Î±=1, the CEGG class becomes the complementary Ggeometric (CGG) family.
Remark 3
Swapping p with q in the ZTG pmf with p+q=1, the cdf of an alternative CEGG (CEGGA) class will be
For Î±=1, the CEGGA class leads to the complementary Ggeometric (CGG) family proposed by Castellares and Lemonte ( 2016 ).
4.2 Complementary exponentiated KumaraswamyGgeometric class (new)
If Î±=1, T[G(z;Ï„)]=K _{ a,b }(z;Ï„) and N be a geometric r.v. with pmf \(\mathbb {P}(N=n)=q\, p^{n1},\, n=1, 2,\dots \), the cdf of the CEKGG class is given by
For Î±=1, the CEKGG class leads to the complementary KGG (CKGG) class.
4.3 Complementary McDonaldGgeometric class (new)
If Î±=1, T[G(z;Ï„)]=M _{ a,b,c }(z;Ï„) and N be a geometric r.v. with pmf \(\mathbb {P}(N=n)=q\, p^{n1},\, n=1, 2,\dots \), the cdf of the complementary McDonaldGgeometric (CMGG) class can follow as
4.4 Complementary betaGgeometric class (new)
If Î±=1, T[G(z;Ï„)]=B _{ a,b }(z;Ï„), and N be a geometric r.v. with pmf \(\mathbb {P}(N=n)=q\, p^{n1},\, n=1, 2,\dots \), the cdf of the complementary betaGgeometric (CBGG) class is given by
4.5 Complementary exponentiated GPoisson class (new)
If T[G(z;Ï„);Î±]=G(z;Ï„)^{Î±} and N be a ZTP r.v. with parameter Î», the cdf of the complementary exponentiated GPoisson (CEGP) can follow as
For Î±=1, the CEGP class leads to the complementary GPoisson (CGP) class.
4.6 Complementary exponentiated KumaraswamyG Poisson class (new)
If T[G(z;Ï„);Î±]=K _{ a,b }(z;Ï„)^{Î±} and N be a ZTP r.v. with parameter Î», the cdf of the complementary exponentiated KumaraswamyG Poisson (CEKGP) class is given by
For Î±=1, the CEKGP class reduces to the complementary KumaraswamyG Poisson (CKGP) class.
4.7 Complementary McDonaldG Poisson class (new)
If Î±=1 and T[G(z;Ï„)]=M _{ a,b,c }(z;Ï„) and N be a ZTP r.v. with parameter Î», the cdf of the complementary McDonaldG Poisson (CMGP) class reduces to
4.8 Complementary beta GPoisson class (new)
If Î±=1, T[G(z);Ï„]=B _{ a,b }(z;Ï„) and N be a ZTP r.v. with parameter Î», the cdf of the complementary beta GPoisson (CBGP) class is given by
4.9 Complementary exponentiated Glogarithmic class (new)
If T[G(z;Ï„);Î±]=G(z;Ï„)^{Î±} and N be a Ln r.v. with pmf given by \(\mathbb {P}(N=n)=(1\phi)^{n}/[n\,\ln \,\phi ],\quad \phi \in (0,1),\quad n=1, 2,\dots \), the cdf of the complementary exponentiated Glogarithmic (CEGLn) class can be expressed as
by noting that \(\sum _{n=1}^{\infty }\,Q^{n}/n=\,\ln (1Q)\). For Î±=1, the CEGLn class becomes the complementary Glogarithmic (CGLn) class.
4.10 Complementary exponentiated KumaraswamyGlogarithmic class (new)
If T[G(z;Ï„);Î±]=K _{ a,b }(z;Ï„)^{Î±} and N be a Ln r.v. with \(\mathbb {P}(N=n)=(1\phi)^{n}/[n\,\ln \,\phi ],\,\phi \in (0,1),\,n=1, 2,\dots \), the cdf of the complementary exponentiated Kumaraswamy Glogarithmic of type 1 (CEKGLn1) class is given by
For Î±=1, the complementary KumaraswamyGlogarithmic (CEKGLn) class becomes the complementary Glogarithmic (CGLn) class.
4.11 Complementary McDonaldGlogarithmic class (new)
If Î±=1, T[G(z;Ï„)]=M _{ a,b,c }(z;Ï„) and N be a Ln r.v. with pmf \(\mathbb {P}(N=n)=(1\phi)^{n}/(n\,\ln \,\phi),\quad \phi \in (0,1),\quad n=1, 2,\dots \), the cdf of the complementary McDonaldGlogarithmic (CMGLn) class can be expressed as
4.12 Complementary betaGlogarithmic class (new)
If Î±=1 and T[G(z;Ï„)]=B _{ a,b }(z;Ï„) and N be a Ln r.v. with pmf \(\mathbb {P}(N=n)=(1\phi)^{n}/(n\,\ln \,\phi),\,\phi \in (0,1),\, n=1, 2,\dots \), the cdf of the complementary betaGlogarithmic (CBGLn) class is given by
4.13 Complementary exponentiated Gbinomial class (new)
If T[G(z;Ï„);Î±]=G(z;Ï„)^{Î±} and N be a ZTBi r.v. with pmf
the cdf of the complementary exponentiated Gbinomial (CEGBi) class is given by
For Î±=1, the CEGBi class becomes the complementary Gbinomial (CGBi) class.
4.14 Complementary exponentiated GNB class (new)
If T[G(z;Ï„);Î±]=G(z;Ï„)^{Î±} and N be a ZTNB r.v. with parameter Î²âˆˆ(0,1) with pmf
Using the binomial expansion \((1w)^{s}1=\sum _{n=1}^{\infty } \binom {s+n1}{n}\, w^{n}\), the cdf of the complementary exponentiated GNB (CEGNB) class reduces to
For Î±=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.
Exponentialgeometric (EG) model. If Xâˆ¼Exponential(Î²) have cdf G(x;Î²)=1âˆ’ exp(âˆ’Î² x), then the cdf of the EG model pioneered by Adamidis and Loukas (1998) (as the first compounded model in the literature) can be determined from the GG class (Section 3.1 with Î±=1) as
Adamidis et al. (2005) also defined an extended EG model.
Generalized EG (GEG) model. If Xâˆ¼Exponential(Î²), Silva et al. (2010) applied the LA1 class to the EG cdf in the last equation and defined the GEG cdf by
where Î³>0 is the power parameter, Î²>0 and pâˆˆ(0,1).
Complementary EG (CEG) model (proposed). If Xâˆ¼Exponential(Î²), then the CEG cdf can be determined from the CEGGA class (Section 4.1 with Î±=1) as
Longterm complementary EG (LCEG) model. If Xâˆ¼Exponential(Î²), then Louzada et al. (2012d) proposed another CEG model called the LCEG model, in a latent complementary risk framework, with cdf given by
where Î²>0 is a scale parameter, Î¸ is a shape parameter and p is the longterm parameter.
Generalized complementary EG (GCEG) model. Bidram and Nadadrajah (2016) applied the LA1 class to the CEGA cdf (given in Section 4.1 with Î±=1) and obtained the GCEG cdf as
where Î³>0 is the power parameter, Î²>0 and pâˆˆ(0,1).
Complementary exponentiatedexponential geometric (CEEG) model. If Xâˆ¼ExponentiatedExponential(Î²,Î±) have cdf G(x;Î²,Î±)=[1âˆ’ exp(âˆ’Î² x)]^{Î±}, then the cdf of the CEEG model, proposed by Louzada et al. (2014a), can be determined from the alternative CEGG class (Section 4.1 with Î±=1) as
ExponentiatedCEG (ECEG) model. Yamachi et al. (2013) generalized the CEG model \(F_{CEG}(z; p, \beta, b)=\frac {p(1\mathrm {e}^{b\,\beta \,z})}{\mathrm {e}^{\beta \,z}(1p)+p}\) introduced by Louzada et al. (2011) by applying the LA1 class and defined the ECEG cdf as
where Î³>0 is the power parameter.
BetaEG (BEG) model. If Xâˆ¼Exponential(Î²), the cdf of the BEG model proposed by Bidram (2012) and Nassar and Nada (2012) can follow from the BGG class (Section 3.4) as
where
Exponentiated EG (EEG) model. The cdf of the EEG model defined by Louzada et al. (2014) can be determined from the EGGA class (Section 3.1) as
RistiÄ‡ and Kundu (2016) proposed the generalized geometric extreme distribution, which is identical to the EEG model given above.
Modified EG (MEG) model. Let Xâˆ¼ModifiedExponential(Î²,Ï€) or the MOexponential with cdf G(x;Î²,Ï€)=1âˆ’ exp(âˆ’Î² x)/[1âˆ’(1âˆ’Ï€) exp(âˆ’Î² x)]. The cdf of the modified EG (MEG) distribution given by Bordbar and Nematollah (2016) can follow from the GG class (Section 3.1 with Î±=1) as
ExponentiatedLL geometric (ELLG) and complementary exponentiatedLL geometric (CELLG) models. If Xâˆ¼ELL(a,b) has cdf \(G(x;a,b,\alpha)=\frac {\alpha \,b}{a} (x/a)^{b1} \left [1+(x/a)^{b}\right ]^{\alpha 1}\), then the cdfs of the ELLG and CELLG models proposed by Mendoza et al. (2016) can be determined from the GG and CGG classes described in Sections 3.1 and 4.1 (with Î±=1), respectively, as
and
Generalized linear failure rategeometric (GLFRG) model. If Xâˆ¼LinearExponential(a,b) have cdf G(x;a,b)=1âˆ’ exp[âˆ’(a x+(b/2)x ^{2})], then the cdf of the GLFRG model proposed by Nadarajah et al. (2014a) can follow from the EGG class (Section 3.1) as
Complementary generalized linear failure rategeometric (CGLFRG) model. If Xâˆ¼LinearExponential(a,b) have cdf G(x;a,b)=1âˆ’ exp[âˆ’(a x+(b/2)x ^{2})], then the CGLFRG cdf defined by Harandi and Alamatsaz (2016) can be determined from the CEGG class (Section 4.1) as
Exponentiated power Lindleygeometric (EPLG) model. If Xâˆ¼ExpPowerLindley(Î±,Î²,Î¸) introduced by Ashour and Eltehiwy (2015) have cdf \(G(x;\alpha, \beta, \theta)=\left [1\left (1+\frac {\theta \,x^{\beta }}{\theta +1}\right)\,\mathrm {e}^{\theta \,x^{\beta }}\right ]^{\alpha }\), the EPLG cdf defined by Alizadeh et al. (2016b) can follow from the GG class (Section 3.1) as
Weibullgeometric (WG) model. If Xâˆ¼Weibull(a,b) have cdf \(G(x; a,b)=1\mathrm {e}^{(b\,x)^{a}}\), then the cdf of the WG model proposed by BarretoSouza et al. (2011) can be expressed from the GG class (Section 3.1 with Î±=1) as
Exponentiated Weibullgeometric (EWG) model. If Xâˆ¼Weibull(a) have cdf \(G(x;a)=1\mathrm {e}^{x^{a}}\), then the EWG cdf given by Chung and Kang (2014) can be obtained from the EGG cdf (Section 3.1) as
BetaWeibullgeometric (BWG) model. If Xâˆ¼Weibull(a,b) have cdf \(G(x;a,b)=1\mathrm {e}^{(b\,x)^{a}}\), then the cdf of the BWG model defined by Bidram et al. (2013) and Cordeiro et al. (2013b) can follow from the BGG cdf (given in Section 3.4) as
where
Complementary exponentiated Weibullgeometric (CEWG) model. If X have Weibull cdf \(G(x;a,b)=1\mathrm {e}^{(b\,x)^{a}}\), the cdf of the CEWG model proposed by Mahmoudi and Shiran (2012a) follows from the CEGG class (Section 4.1) as
Complementary Weibullgeometric (CWG) model. If Xâˆ¼Weibull(a,b) have cdf \(G(x;a,b)=1\mathrm {e}^{(b\,x)^{a}}\), the cdf of the CWG model proposed by Tojeiro et al. (2014) follows from the alternative CGG class given in Section 4.1 (Î±=1) as
Modified Weibullgeometric (MWG) model. If Xâˆ¼ModifiedWeibull(a,b,Î´) introduced by Sarhan and Zaindin (2009) have cdf G(x;a,b,Î´)=1âˆ’ exp[âˆ’(a x+b x ^{Î´})], then the cdf of the MWG model defined by Wang and Elbatal (2015) can be determined from the GG class (Section 3.1, Î±=1 and all parameters positive) as
Kummerbeta Weibullgeometric (KBWG) model. Ashour and Wahed (2014) extended the WG model to the KBWG model. The KBWG cdf is given by
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.
Additive Weibullgeometric (AWG) model. If Xâˆ¼AddiveWeibull(a,b,Î³) introduced by Cordeiro et al. (2014c) have cdf \(G(x;a, b, \gamma)= 1\mathrm {e}^{(a x+b x^{\gamma })},\quad x, a, b>0\) and Î³âˆˆ(0,âˆž)âˆ–{1}, then the cdf of the AWG model defined by Elbatal et al. (2016) can be obtained from the GG class (Section 3.1, Î±=1) as
Lindleygeometric (LG) model. If Xâˆ¼Lindley(Î¸) have cdf \(G(x;\theta)=1\left (1+\frac {\theta \,x}{\theta +1}\right)\,\mathrm {e}^{\theta \,x}\) (for x>0, Î¸>0), then the cdf of the LG model proposed by Zakerzadeh and Mahmoudi (2013) can follow from the GG class (Section 3.1, Î±=1) as
Exponentiated Lindleygeometric (ELG) model. If Xâˆ¼Lindley(Î¸) have cdf \(G(x;\theta)=1\left (1+\frac {\theta \,x}{\theta +1}\right)\,\mathrm {e}^{\theta \,x}\) (for x>0, Î¸>0), then the cdf of the ELG model proposed by Wang (2013) follows from the EGG class (Section 3.1) as
5.2 Compounded models based on Poisson distribution
ExponentialPoisson (EP). If Xâˆ¼Exponential(Î²), then the cdf of the EP model proposed by KuÅŸ (2007) can be obtained from the GP class defined in Section 3.5 (with Î±=1) as
For some structural properties and applications of the EP model, the reader is referred to KuÅŸ (2007) and Cancho et al. (2011a).
Generalized EP (GEP) model. If Xâˆ¼Exponential(Î²), BarretoSouza and CribariNeto (2009) obtained the GEP cdf by applying the LA1 class to the EP cdf given in the last equation
where Î³>0 is the power parameter, Î²>0 and pâˆˆ(0,1).
Complementary EP (CEP) model. If Xâˆ¼Exponential(Î²), the cdf of the CEP model, proposed by Cancho et al. (2011a) and Rezaei and Tahmasbi (2012), can be determined from the CGP class (Section 4.5, Î±=1) as
Modified EP (MEP) model. If Xâˆ¼ModifiedExponential(Î²) or MOexponential with cdf \(G(x;\beta)=\frac {1\mathrm {e}^{\beta \,x}}{1(1p)\mathrm {e}^{\beta \,x}}\), then the cdf of the MEP model, proposed by Preda et al. (2011), can follow from the GP class (Section 3.9, Î±=1) as
Exponentiated EP (EEP) model. If Xâˆ¼Exponential(Î²), then the cdf of the EEP model given by RistiÄ‡ and Nadarajah (2014) can be obtained from the EGP class defined in Section 3.5 as
WeibullPoisson (WP) model. If Xâˆ¼Weibull(a) have cdf \(G(x;a,b)=1\mathrm {e}^{bx^{a}}\), then the cdf of the WP model proposed by Lu and Shi (2012) can follow from the GP class (Section 3.5, Î±=1) as
Hemmati et al. (2011) and Bereta et al. (2011) also obtained the WP or PoissonWeibull (PW) models from the Weibull cdf G(y;a,b)=1âˆ’ exp{âˆ’(b y)^{a}}. The cdf of their WP model is given by
BetaWeibull Poisson (BWP) model. If Xâˆ¼Weibull(a,b) have cdf \(G(x;a,b)=1\mathrm {e}^{bx^{a}}\), then the cdf of the BWP model, proposed by Percontini et al. (2013a), can follow from the BGP class defined in Section 3.8 as
where
Complementary modified WeibullPoisson (CMWP) model. If Xâˆ¼ModifiedWeibull(a,b,Î´) introduced by Sarhan and Zaindin (2009) have cdf G(x;a,b,Î´)=1âˆ’ exp[âˆ’(a x+b x ^{Î´})], then the cdf of the CMWP model proposed by Ghorbani et al. (2014) can be obtained from the GP class presented in Section 3.5 (with Î±=1) as
Complementary exponentiated WeibullPoisson (CEWP) model. If Xâˆ¼Weibull(a,b) have cdf \(G(x;a,b)=1\mathrm {e}^{(bx)^{a}}\), then the cdf of the CEWP model defined by Mahmoudi and Sepahdar (2013) can follow from the EGP class (see Section 3.5) as
Loglogistic generalized Weibull Poisson (LLGWP) model. Let Xâˆ¼Weibull(a,b) have cdf \(G(x;a,b)=1\mathrm {e}^{bx^{a}}\). Oluyede et al. (2016a) first defined the loglogistic generalized Weibull (LLGW) cdf by \(F_{LLGW}=1\left [1+(x/s)^{c}\right ]^{1}\, \mathrm {e}^{bx^{a}}\) from the generalized extended Weibull family G(x;Î¸,Ïˆ)=1âˆ’B(x;Ïˆ) exp{âˆ’Î± H(x;Î¸)}, where Î¸ and Ïˆ are both vectors, by taking H(x;Î¸)=x ^{b} and B(x;Ïˆ)=1âˆ’[1+(x/s)^{c}]. Then, the cdf of the LLGWP model proposed by Oluyede et al. (2016c) can follow from the GP class (Section 3.5, Î±=1) as
Laimodified WeibullPoisson (LaiMWP) model. If Xâˆ¼ModifiedWeibull(a,b,Î´) introduced by Lai et al. (2003) have cdf G(x;a,b,Î´)=1âˆ’ exp(âˆ’a x ^{Î´} e^{bx}),a,b,Î´>0, then the cdf of the LaiMWP model proposed by Delgarm and Zadkarami (2015) can be obtained from the GP class (Section 3.5, Î±=1) as
Exponentiated LomaxPoisson (ELoP) model. If Xâˆ¼Lomax(a,b) have cdf G(x;a,b)=1âˆ’(1+b x)^{âˆ’a}, then the cdf of the ELoP model proposed Ramos et al. (2013) can be determined from the EGP class (Section 3.5) as
Complementary PoissonLomax (CPLo) model. Let Xâˆ¼Lomax(a,b) have cdf G(x;a,b)=1âˆ’(1+b x)^{âˆ’a}, then the cdf of the CLoP model proposed AlZahrani and Sagor (2014) can follow from the CGP class (Section 4.5, Î±=1) as
LindleyPoisson (LP) model. Let Xâˆ¼Lindley(Î¸) have cdf \(G(x;\theta)=1\left (1+\frac {\theta \,x}{\theta +1}\right)\, \mathrm {e}^{\theta \,x},\, x, \theta >0\). The cdf of the LP model introduced by Gui et al. (2014) can be determined from the GP class (Section 3.5, Î±=1) as
Complementary extended LindleyPoisson (CExtLP) model. If Xâˆ¼ExtLindley(Î¸,Î´,Î²) proposed by Bakouch et al. (2012a) have cdf \(G(x;\theta, \delta, \beta)=1\left (\frac {1+\theta +\theta \,x}{\theta +1}\right)^{\delta }\, \mathrm {e}^{(\theta \,x)^{\beta }}\), the cdf of the CExtLP model introduced by Pararai et al. (2015b) can be determined from the CGP class (Section 4.5, Î±=1) as
Poisson BirnbaumSaunders (PBS) model. Let Xâˆ¼BS(a,b) have cdf \(G(x;a,b)=\Phi \left [\left (1/a\right)+\left (\sqrt {x/b}\sqrt {b/x}\right)\right ],\, x, a, b>0\), then the cdf of the PBS model, proposed by Hashimoto et al. (2014) in terms of longterm survivors, can be obtained from the GP class (Section 3.5, Î±=1) as
ExponentiatedBurr XII Poisson (EBP) model. Let Xâˆ¼Burr(c,k,s) have cdf G(x;c,k,s)=1âˆ’[1+(x/s)^{c}]^{âˆ’k}, x,c,k,s>0, then the cdf of the EBP model introduced by daSilva et al. (2015b) can be determined from the EGP class (Section 3.5) as
Complementary Burr III Poisson (CB3P) model Let Xâˆ¼BurrIII(c,k) have cdf G(x;c,k)=(1+x ^{âˆ’c})^{âˆ’k}, x,c,k>0, then the cdf of the CB3P model defined by Hassan et al. (2015) can follow from the CGP class (Section 4.5, Î±=1) as
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.
Poisson generalized linear failure rate (PGLFR) model. Let Xâˆ¼GLFR(a,b,Î±) introduced by Sarhan and Kundu (2009) have cdf G(x;a,b,Î±)=(1âˆ’ exp[âˆ’a xâˆ’(b/2)x ^{2}])^{Î±}, x,a,b,Î±>0, then the cdf of the PGLFR model defined by Cordeiro et al. (2015b) can be obtained from the EGP class (Section 3.5) as
Complementary failure rate Poisson (CLFRP) model. Let Xâˆ¼LFR(a,b,Î±), then the cdf of the CLFRP model defined by Gitifar et al. (2016) can follow from the CEGP class (Section 4.5, Î±=1) as
Complementary exponentiated power LindleyPoisson (CEPLP) model. If Xâˆ¼ExpPowerLindley(Î±,Î²,Î¸) introduced by WarahenaLiyanage and Pararai (2015b) have cdf \(G(x;\alpha, \beta, \theta)=\left [1\left (1+\frac {\theta \,x^{\beta }}{\theta +1}\right)\,\mathrm {e}^{\theta \,x^{\beta }}\right ]^{\alpha }\) (for x>0, Î±,Î²,Î¸>0), then the EPLG cdf defined by Pararai et al. (2016) can follow from the CGP class (Section 4.5) as
5.3 Compounded models based on the logarithmic distribution
The following compounded models have been reported in the literature from the logarithmic discrete model.
Exponentiallogarithmic (ELn). If Xâˆ¼Exponential(Î²) have cdf G(x;Î²)=1âˆ’ exp(âˆ’Î² x), then the cdf of the exponentiallogarithmic (ELn) model proposed by Tahmasbi and Rezaei (2008) can be obtained from the GLn class (Section 3.9, Î±=1) as
Generalized ELn. Pappas et al. (2015) applied the LA1 class to the ELn cdf given in Eq. (8), and obtained the cdf of the generalized ELn (GELn) distribution as
where Î³>0 is the power parameter.
Chenlogarithmic (ChLn). If Xâˆ¼Chen(Î³,Î²) have cdf \(G(x;\gamma, \beta)=\mathrm {e}^{\gamma (\mathrm {e}^{x^{\beta }}1)}\phantom {\dot {i}\!}\) introduced by Chen (2000), then the cdf of the Chenlogarithmic (ChLn) model proposed by Pappas et al. (2011) can be determined from the GLn class (Section 3.9 with Î±=1) as
Complementary exponentiated Weibulllogarithmic (CEWLn) model. Let Xâˆ¼EW(Î²,Î´,Î±) have cdf \(G(x; \beta, \delta, \gamma)=\alpha \,\delta \,\beta ^{\delta }\,y^{\delta }\,\mathrm {e}^{(\beta \,y)^{\delta 1}} [1\mathrm {e}^{(\beta \,y)^{\delta }}]^{\alpha 1}\phantom {\dot {i}\!}\), then the cdf of the complementary exponentiated Weibulllogarithmic (CEWLn) model introduced by Mahmoudi et al. (2014) can follow from the cdf of the CGLn class given in Section 4.9 as
where Ï•âˆˆ(0,1), Î±>0 and Î´>0 are shape parameters and Î²>0 is a scale parameter.
Lomaxlogarithmic (LxLn) model. Let Xâˆ¼Lomax(a,b) have cdf G(x;a,b)=(1+b x)^{âˆ’a}, then the cdf of the Lomaxlogarithmic (LxLn) model given by AlZahrani and Sagor (2015) can be obtained from the GLn class given in Section 3.9 with Î±=1 as
Exponentiated Poissonlogarithmic (EPoLn) model. Let Xâˆ¼ExpPoisson(Î¸,Î²) have cdf \(G(x;a,b)=1\frac {1\exp \{\theta \, \mathrm {e}^{\beta \,x}\}}{1\mathrm {e}^{\theta }}\), then the cdf of the exponentiated Poissonlogarithmic (EPoLn) model given by Fioruci et al. (2016) can follow from the alternative GLn cdf (Section 3.9 with Î±=1) as
Other compounded models
(i) Extended Weibull power series (EWPS) models. Let T[G(y;Ï„);Î±]=1âˆ’e^{âˆ’Î±H(y;Ï„)} be the extended Weibull family, where x>0, Î±>0 and H(y;Ï„) is a nonnegative monotonically increasing function depending on a parameter vector Ï„. Much more than twenty known lifetime model cdfs can be expressed in this form. Further, let N be a discrete random variable following a power series (PS) distribution (truncated at zero) with pmf
where a _{ n } depends only on n, \(C(\theta) = \sum _{n=1}^{\infty }a_{n} \,\theta ^{n}\) and Î¸>0 is such that C(Î¸) is finite. Equation (9) summarizes some power series distributions (truncated at zero) such as the Poisson, logarithmic and geometric distributions, where C(Î¸) is equal to (e^{Î¸}âˆ’1), âˆ’ log(1âˆ’Î¸) and Î¸(1âˆ’Î¸)^{âˆ’1}, respectively.
Silva et al. (2013) defined the EWPS cdf obtained under the current setup from Eq. (4) as
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 Ã— 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.
(ii) Complementary extended Weibull power series (CEWPS) models. The CEWPS class follows by taking T[G(y;Ï„);Î±]=1âˆ’e^{âˆ’Î±H(y;Ï„)} and the PS distribution for the pmf. So, the cdf of the CEWPS class can be obtained from Eq. (5) as
The CEWPS class can arise in parallel systems with identical components, which appear in many industrial and biological applications.
(iii) Exponentialgeometric range (maxmin) model (New). Let N be the ZTG r.v. with pmf of type 1 \(\mathbb {P}(N=n)=q\,p^{n1},\, n=1, 2, \ldots \) and define the range X _{ R }= max{Y _{1}â€¦,Y _{ N }p}âˆ’ min{Y _{1}â€¦,Y _{ N }p}. The unconditional cdf of the Ggeometric range (R= maxâˆ’ min) can be obtained as
If Yâˆ¼E x p(Î²) have density f _{ YÎ² }(y)=Î²e^{âˆ’Î²y} and Zpâˆ¼G e o m e t r i c(p) have pmf \(\mathbb {P}(N=n)=q\,p^{n1},\,n=0,1, \ldots \), Shahsanaei et al. (2012) proposed the unconditional cdf of the exponentialgeometric range (EGR) model as
The EGR distribution is useful for modeling the time between the first failure to the last failure.
(iv) Exponentialgeneralized Poisson (EGP) model. Let N have the zero truncated generalized Poisson (ZTGP) distribution with parameters Î» and Î± and pmf
Then, Gupta et al. (2014) proposed the EGP cdf given by
where g=âˆ’Î± e^{âˆ’(Î±+Î²y)} and W(Â·) is the Lambert W function defined by W(x) e^{W(x)}=x. The above cdf can be derived from the generating function of the ZTGP distribution \(M_{N}^{\ast }(t)=\frac {M_{N}(t)P(N=0)}{1P(N=0)}=\frac {1\mathrm {e}^{(\lambda /\alpha) [W(g)+\alpha ]}\mathrm {e}^{\lambda }}{1\mathrm {e}^{\lambda }}\), which is equivalent to the survival function 1âˆ’F(t;Î±,Î²,Î»).
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 (PopoviÄ‡ et al.2015), geometric exponential Poisson (Nadarajah et al.2013a) and additive Weibull (Xie and Lai1995) distributions.
(i) ExponentialWeibull model. Let W and T be independent random variables with exponential and Weibull distributions characterizing two series subsystem lifetimes, whose cdfs are G _{ W }(z)=1âˆ’ exp(âˆ’Î± z) and G _{ T }(t)=1âˆ’ exp(âˆ’Î² t ^{Î³}), respectively. Here, Î±>0 and Î³>0 are shape parameters and Î²>0 is a scale parameter. The distribution of the random variable X= min{W,T} is called the exponentialWeibull (EW) distribution, whose cdf is given by
We consider that Î±>0, Î²>0 and Î³âˆˆ(0,âˆž)âˆ–{1}, which gives identifiability to the model. The mathematical properties of the EW model were investigated by Cordeiro et al. (2014c).
(ii) Generalized exponentialexponential (GEE) model. Gupta and Kundu (1999) introduced the twoparameter generalized exponential (GE) distribution with cdf given by G(x;Î±,Î»)=(1âˆ’e^{âˆ’Î»x})^{Î±}, where x>0,Î±>0,Î»>0. For a given Î±, let Y be a random variable with the GE cdf with parameters Î±>0 and Î»>0. Let Î± be a random variable following an exponential distribution with parameter Î¸>0. Then, PopoviÄ‡ et al. (2015) defined the GEE cdf (for x>0) by
(iii) Exponentiated geometric GPoisson (EGGP) model (New). If Nâˆ¼Z T G(p) and Mâˆ¼Z T P(Î»), then the EGGP family can be determined by inserting the EGP cdf (Section 3.9) in the cdf of the EGG class (Section 3.1) as
If Î±=1 and Xâˆ¼E x p(Î²) 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)=1âˆ’ exp(âˆ’a t ^{b}âˆ’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.
(i) If the cdf of the GP class defined in Section 3.5 with Î±=1 is inserted into the cdf of GammaG class, proposed by Zografos and Balakrishnan (2009), then the cdf of new class (GaGP) is obtained as
where \(\Gamma (a)=\int _{0}^{\infty } t^{a1}\,\mathrm {e}^{t}\, dt\) and \(\Gamma {(a,z)}=\int _{z}^{\infty } t^{a1}\,\mathrm {e}^{t}\, dt\) are the gamma and upper incomplete gamma functions, respectively, and \(\bar {G}_{GP}(y)=1G_{GP}(y).\)
One special model of the GaGP class, that is, the gammaWeibullPoisson (GaWP) was studied by Percontini et al. (2014), whose cdf is given by
(ii) The cdf of the new KumaraswamyGP (KGP) class can be obtained by inducting the cdf of the CGP model given in Section 4.5 with Î±=1 into the cdf of the KumaraswamyG class, pioneered by Cordeiro and deCastro (2011), as
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:
(i) Transmuted Weibullgeometric (TWG). Merovci and Elbatal (2014b) extended the WG model introduced by BarretoSouza et al. (2011) (given in Section 5) with the transmutedG family and defined the cdf of the TWG model by
(ii) Exponentiated transmutedWeibull geometric (ETWG) model. If Xâˆ¼Weibull(a,b) have cdf \(G(x;a,b)=1\mathrm {e}^{(x/b)^{a}}\), Aryal and Tsokos (2011) defined the cdf of the transmutedWeibull distribution as
The cdf of the ETWG model proposed by AlZahrani et al. (2015) can follow from the EGG cdf defined in Section 3.1 as
(iii) Transmuted complementary Weibullgeometric (TCWG) model. Afify et al. (2014) applied the LA1 class to the cdf of the Tojeiro et al.â€™s CWG model given in Section 5, and obtained the cdf of the TCWG model as
(iv) Transmuted complementary exponentiated Weibullgeometric (TCEWG) model. If Xâˆ¼ExpWeibull(a,b,Î±) have cdf \(G(x;\beta)=\left (1\mathrm {e}^{(b\,x)^{a}}\right)^{\alpha }\), Saboor et al. (2016) used the CEWG cdf given in Section 5 and obtained the cdf of the TCEWG model as
(v) Transmuted Lindleygeometric (TLG) model. Merovci and Elbatal (2014a) extended the LG model (described in Section 5) by proposing the TLG cdf given by
where Î· and Î¸>0 are transmuted and scale parameters, respectively, and pâˆˆ(0,1).
(vi) Transmuted exponentialWeibulll (TEW) model. Saboor et al. (2015) recently extended the exponentiatedWeibull model originally proposed by Cordeiro et al. (2014c). The TEW cdf is given by
where Î±>0, Î²>0 and Î³âˆˆ(0,âˆž)âˆ–{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 CramÃ©râ€“von Mises (W ^{âˆ—}) and AndersonDarling (A ^{âˆ—}) 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.
In the literature, two formulations of the cure rate models have received increased attention, namely, the mixture cure rate model (Boag1949; Berkson and Gage1952) and promotion time (or accelerated) cure rate model (Yakovlev and Tsodikov1996; Chen et al.1999). Perhaps, the mixture cure rate models has received increased popularity. In mixture model, it is assumed that there is a fraction 1âˆ’Ï€ _{0} of susceptible individuals and, hence, a proportion Ï€ _{0} of cured individuals. The survivor function for the entire population is
where S _{0} denotes the survival function for the noncured group in the population. So, the flexibility and performance of the population model depends on the choice of S _{0}. Rodrigues et al. (2009a) unified both formulations considering the negative binomial cure rate model. The proper modeling of cure rate models reported in literature is with covariates, which have been proved very useful in getting improved results for the estimated cure rate. Some selected cure rate models are reported in Table 2. Further detail is beyond the scope of the paper.
9.3 Inference for cure fraction models
Suppose that the timetoevent is not completely observed and may be subject to right censoring. Let C _{ i } denote the censoring time. We observe that Y _{ i }= min(T _{ i },C _{ i }) and Î´ _{ i }=I(T _{ i }<C _{ i }) is such that Î´ _{ i }=1, if T _{ i } is a timetoevent and Î´ _{ i }=0, if it is right censored, i=1,â€¦,n. Let Î˜ denote the parameter vector of the distribution of the timetoevent. From n pairs of times and censoring indicators D a t a=(y _{1},Î´ _{1}),â€¦,(y _{1},Î´ _{1}), the corresponding likelihood function under uninformative censoring is
where f _{ CRM }(D a t a,Î˜) and F _{ CRM }(D a t a,Î˜) are the density and cdf of a cure rate model (CRM), respectively.
Maximum likelihood estimation of the parameter vector Î˜ is carried out through maximization of the loglikelihood function â„“(D a t a,Î˜)= logL(D a t a,Î˜). Details for the maximization methods are discussed in Section 9.1. Under suitable regularity conditions, it can be shown that the asymptotic distribution of the maximum likelihood estimator (MLE) \(\widehat {\boldsymbol {\Theta }}\) is normal with mean vector Î˜ and covariance matrix \(\boldsymbol {\sum }(\widehat {\boldsymbol {\Theta }})\) given by
evaluated at \(\boldsymbol {\Theta }=\widehat {\boldsymbol {\Theta }}\). The required second derivatives are computed numerically.
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;Ï„);Î±] 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.
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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.
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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.
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The authors declare that they have no competing interests.
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Tahir, M.H., Cordeiro, G.M. Compounding of distributions: a survey and new generalized classes. J Stat Distrib App 3, 13 (2016). https://doi.org/10.1186/s4048801600521
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DOI: https://doi.org/10.1186/s4048801600521