Poisson Generated Family of Distributions: A Review

Abstract

The present article represents a survey on Poisson generated family of distributions. Based on this family of distribution, several transformations and distributions have been proposed. Out of which, some of them are proposed by referencing it, and some are independent. The family can be proposed by using the compounding concept of zero truncated Poisson distribution with any other model or family of distributions. Here, we provide a complete survey on this family of distributions and list the contributory related research works. We also address 12 power series distributions, 77 distributions based on the Poisson family of distribution, and 23 distributions, based on different ten transformation methods based on this family of distribution. These numbers show the importance of the Poisson family of distribution.

1 Introduction

In statistical literature, we suppose that every real phenomenon is governed by some lifetime model. If we know the model, we can completely specify our problem or phenomenon as various lifetime models have been developed for this purpose. Poisson distribution is one of the famous models that also provide a family of distribution. By using that family, a number of lifetime models have been proposed and studied their properties by several authors.

The Poisson family of distribution (PFD) can be constructed by using the concept of compounding. In the compounding method, there are two different ways available; one is by using zero truncated power series distribution and others by using zero truncated Poisson distribution directly with other continuous lifetime models. There are two basic reasons behind using zero truncated Poisson distribution or power series distribution. The first one is that in a real dataset, there are negligible instances of the value zero, which compels one to consider the value zero to be excluded, and the second one is that the compounding process is based on the number of complementary risk for failures of components. Hence, it is more appealing that this number must be greater than or equal to one. In this paper, we provide a detailed study of both approaches for constructing PFD. The main idea of compounding is that the lifetime of a system with N (a discrete random variable) components and the failure time of i^th component T_i (say), independent of N, follow some continuous lifetime distribution. Then the maximum or minimum time of failures of components of the system depending on the condition whether they are parallel or in series, respectively. One of the base paper that in-light in the field of compounding was proposed by Adamidis and Loukas (1998).

PFDs are more suitable in case of complementary and competitive risk. In these scenarios, information about a particular factor that is responsible for the death or failure is not available, and only the lifetime of maximum or minimum among all the elements in risk are available. These phenomenons frequently occur in reliability, data modeling, biological systems, the health care field, and actuarial science. Also, the Poisson generated family of distributions are more flexible than the existing baseline models and are capable of fitting to the wide variety of skewed data.

There are also so many transformation techniques that are available in the literature to generate new lifetime models. And various authors proposed a number of lifetime distributions, studied their corresponding statistical properties and applications in the real-life scenario by utilizing these techniques. Some of these techniques are directly related to PFD, but many of the time, its rule in developing models are ignored. This paper also discussed how these techniques are related to PFD.

The main focus of the paper is two-fold; first, we provide an updated list of Poisson generated family of distribution that will be helpful for researchers in the field of distribution theory. Secondly, it provides different transformation techniques that are actually somehow related to PFD, that also helpful for generating new lifetime models.

1.1 Genesis for the Poisson Generated Family of Distribution

Let N be a random variable denoting the number of complementary risk related to the occurrence of an event of interest. Further, we assume that N has a zero truncated Poisson distribution with probability mass function (PMF) given by;

$$ P[N=n]=\frac{e^{-\theta}\theta^{n}}{n! (1-e^{-\theta})}\qquad\qquad n=1,2,\cdots,;\theta>0. $$

(1)

Now, let T_i(i = 1,2,⋯ ,) denotes the time to failure of event due to i^th system, independent of N follow some distribution having probability density function (PDF) g(t) and cumulative distribution function (CDF) G(t). Now, let the maximum time of system is denoted by X = max(T₁,T₂,⋯ ,T_N). Then the conditional distribution of X given N = n is

$$ f(x|N=n)=ng(x)[G(x)]^{n-1} $$

and the marginal distribution of X is,

$$ \begin{array}{ll} f(x)=&{\sum}_{n=1}^{\infty}ng(x)[G(x)]^{n-1}\frac{e^{-\theta}\theta^{n}}{n!(1-e^{-\theta})}\\ =&\frac{\theta g(x)e^{\theta G(x)}}{e^{\theta}-1} \end{array} $$

(2)

and the corresponding CDF is;

$$ F(x)=\frac{e^{\theta G(x)}-1}{e^{\theta}-1} \qquad x,\theta>0. $$

(3)

Also, if we do the same for X = min(T₁,T₂,⋯ ,T_N). Then the marginal distribution of X is,

$$ f(x)=\frac{\theta g(x)e^{-\theta G(x)}}{1-e^{-\theta}} \qquad x,\theta>0. $$

(4)

and the corresponding CDF is;

$$ F(x)=\frac{1-e^{-\theta G(x)}}{1-e^{-\theta}} \qquad x,\theta>0. $$

(5)

Then the random variable X is called Poisson generated family of distribution if its CDF is given as in equation Eqs. 3 or 5.

These two transformation can also be converted to each other by considering scale parameter 𝜃 = −𝜃 in other. The form given in the Eq. 5 was proposed by Chen et al. (1999) in context of Bayesian analysis model. Cooner et al. (2007) also used this model for flexible cure rate model under latent activation schemes and called first activation scheme (for minimum Eq. 5 case) and last activation scheme (for maximum Eq. 3 case). Karlis (2009) also used this model.

This paper aims to discuss the role of Poisson generated family of distribution in the field of lifetime models because it generates a lot of distribution functions as well as transformation methods. Tahir and Nadarajah (2015) and Tahir and Cordeiro (2016) provide a surveys on the developments of continuous univariate distributions.

The rest of the paper follows as; Section 2 deals with the power series distribution method for generating Poisson family of distribution, Section 3, deals with compounding technique for generating Poisson family of distributions. This section is divided into two subsections, the first Subsection 3.1, gives the details of Poisson family of distribution for parallel components or in case of maximum, while other Subsection 3.2, gives the details for series components or in case of minimum. Section 4 deals with the different transformation methods based on the Poisson family of distribution. Section 5 deals with real data application for some considered models. The conclusion of the whole paper is summarized in Section 6.

2 Power Series Class of Distribution

One of the important methods for constructing PFD is using power series distribution (PSD). The basic idea of PSD is that, let X be a random variable having some distribution function G(x). Further, let for given N; $X_{1}, X_{2}, \dots , X_{N}$ be independent and identical (iid) random variable having the same distribution function. Here, the number of iid random variable is suppose to be discrete random variable following zero truncated power series distribution, with PMF given as

$$ P(N=n)=\frac{a_{n} \theta^{n}}{C(\theta)},\qquad \theta>0, n=1, 2, \dots $$

where a_n > 0, depends on n only and $C(\theta )={\sum }_{n=1}^{\infty }a_{n} \theta ^{n}$, such that C(𝜃) is finite. Now, if we consider the distribution of maximum i.e. $X=X_{(n)}=\max \limits _{1\le i\le N}X_{i}$ then the conditional distribution of X_(n)|N = n is Gⁿ(x) and in this way the marginal distribution of X_(n) is

$$ F(x)={\sum}_{n=1}^{\infty}\frac{a_{n} \theta^{n}}{C(\theta)}G^{n}(x)= \frac{C(\theta G(x))}{C(\theta)}. $$

(6)

And if, we consider the distribution of minimum i.e. $X=X_{(1)}=\min \limits _{1\le i\le N}X_{i}$ then the conditional distribution of X₍₁₎|N = n is 1 − [1 − G(x)]ⁿ and in this way the marginal distribution of X₍₁₎ is

$$ F(x)={\sum}_{n=1}^{\infty}\frac{a_{n} \theta^{n}}{C(\theta)}(1-[1-G(x)]^{n})= 1-\frac{C(\theta [1-G(x)])}{C(\theta)}. $$

(7)

Now, for the different form of G(x) and distributional nature of iid random variable (whether taken as minimum of maximum), different PSD can be generated. The particular choice of a_n and C(𝜃), one can generate different PSD. For example, if we consider $a_{n}=\frac {1}{n!}$ and C(𝜃) = e^𝜃 − 1, one can generate zero truncated Poisson power series distribution. Some of the related work are summarised below:

Chahkandi and Ganjali (2009) proposed exponential power series (EPS) class of distribution by considering the exponential model as a baseline. EPS was proposed on power series distribution by using the concept of truncation at zero, by the method discussed in Eq. 7. The general form of EPS class of distribution is

$$ F(x)=1-\frac{C({\theta} e^{-{\beta} x})}{C({\theta})}; \quad \beta, {\theta}>0 $$

and particular choice for C(𝜃) = e^𝜃 − 1, gives the Poisson family of distribution as given in Eq. 5 and in similar fashion if we take C(𝜃) = 1 − e^−𝜃 provide the complementary PFD as given in Eq. 3. EPS is generalization of some distributions proposed by Adamidis and Loukas (1998), Kuş (2007), Tahmasbi and Rezaei (2008), and Barreto-Souza and Bakouch (2013).

Morais and Barreto-Souza (2011) proposed Weibull power series (WPS) family of distribution by considering $G(x)=1-e^{-{\beta x}^{\alpha }}$ as Weibull distribution in Eq. 7. The general form of WPS is

$$ F(x)=1-\frac{C(\theta e^{-{\beta x}^{\alpha}})}{C({\theta})}; \qquad x, \alpha, \beta, \theta>0. $$

The EPS can also be obtained from WPS family of distribution.

Latter on Mahmoudi and Jafari (2012) generalised the EPS family of distribution and proposed generalised exponential power series (GEPS) class of distribution by considering G(x) = (1 − e^−βx)^α, as generalised exponential distribution in both the concept discussed in Eqs. 6 and 7. The form of GEPS is

$$ F(x)=\frac{C(\theta(1-e^{-\beta x})^{\alpha})}{C(\theta)}; \qquad \alpha, \beta, \theta>0 $$

and

$$ F(x)=1-\frac{C(\theta[1-(1-e^{-\beta x})^{\alpha}])}{C(\theta)}; \qquad \alpha, \beta, \theta>0. $$

The above general form was defined for marginal distribution of maximum of given random variable, in the same way it will define for minimum.

Mahmoudi and Shiran (2012) proposed exponentiated Weibull power series distribution, by using $G(x)=(1-e^{-\beta x^{\alpha }})^{\gamma }$, as exponentiated Weibull in both the concept discussed in Eqs. 6 and 7 as;

$$ F(x)=\frac{C(\theta (1-e^{-\beta x^{\alpha}})^{\gamma})}{C(\theta)}; \qquad \beta, \theta, \alpha, \beta, \gamma>0 $$

and

$$ F(x)=1-\frac{C(\theta [1- (1-e^{-\beta x^{\alpha}})^{\gamma}])}{C(\theta)}; \qquad \beta, \theta, \alpha, \beta, \gamma>0. $$

For particular choice of γ = 1, it reduces to Morais and Barreto-Souza (2011). This PSD is generalization of all above PSD.

Flores et al. (2013) proposed complementary exponential power series (CEPS) distribution by using the similar idea as Chahkandi and Ganjali (2009) and Morais and Barreto-Souza (2011). The CEPS concept is different from EPS as it utilizes the maximum concept instead of minimum, as given in Eq. 6 with the CDF

$$ F(x)=\frac{C(\theta [1-e^{-\beta x}])}{C(\theta)}; \qquad \beta, \theta>0. $$

Silva et al. (2013) proposed extended Weibull power series (EWPS) distribution. This model is zero truncated power series distribution on extended Weibull distribution, and also proposed Pareto Poisson distribution by utilizing the similar concept as given in Eq. 7. Actually, EWPS generalizes the WPS class of distributions and in this way, extends the exponential power series class of distributions. The CDF of EWPS distribution is

$$ F(x)=1-\frac{C(\theta e^{-\alpha H(x,\xi)}) }{C(\theta)} \qquad x,\theta, \alpha>0. $$

Now, if H(x,ξ) = x, converted to Chahkandi and Ganjali (2009) and if H(x,ξ) = x^α, it reduced to Morais and Barreto-Souza (2011).

Leahu et al. (2013) also proposed Poisson generated family of distribution by using power series distribution. This family is obtained as the maximum and minimum sequence of iid random variables and the numbers of components are of power series random variable. The both form was same as given in Eqs. 6 and 7 respectively. The proposed general family of distributions are same as given in Eqs. 3 and 5 respectively. Latter, Leahu et al. (2014) introduced two compound families, namely the max-Erlang Power series distribution and min-Erlang Power series distribution and in this way max-Erlang Poisson and min-Erlang Poisson distribution also proposed. The max-Erlang Power series distribution is

$$ F(x)=\frac{C(\theta[1-e^{-\lambda x}{\sum}_{i=0}^{k-1}\frac{(\lambda x)^{i}}{i!}] )}{C(\theta)}; \qquad \theta, \lambda>0 $$

and min-Erlang Power series distribution is

$$ F(x)=1-\frac{C(\theta[e^{-\lambda x}{\sum}_{i=0}^{k-1}\frac{(\lambda x)^{i}}{i!}] )}{C(\theta)}; \qquad \theta, \lambda>0. $$

Silva and Cordeiro (2015) proposed the Burr XII power series (BXIIPS) class of distribution by compounding the Burr type XII and power series distributions in Eq. 7. The method of compounding is the same as Adamidis and Loukas (1998). The general family is

$$ F(x)=1-\frac{C(\theta (1+x^{c})^{-k} )}{C(\theta)}; \qquad c, \theta, k>0 $$

This family incorporates sub-models as exponential power series and Weibull power series distributions in addition to those which arise as special models of the Burr type XII distribution. They also proposed Burr XII Poisson distribution.

Oluyede et al. (2016a) proposed Dagum power series distribution. This can be obtained by taking Dagum distribution as G(x) in Eq. 6. The Dagum PSD is

$$ F(x)=\frac{C(\theta[1+\lambda x^{-\alpha}]^{-\beta})}{C(\theta)} \qquad x,\theta, \beta, \alpha, \lambda >0. $$

Mahmoudi and Jafari (2017) proposed linear failure rate PSD by compounding linear failure rate distribution with power series distribution, when the components are in series i.e. using the minimum concept as given in Eq. 7. The general family is given as

$$ F(x)=1-\frac{C(\theta[e^{-\beta x-\frac{\alpha}{2}x^{2}}])}{C(\theta)}; \qquad \alpha, \beta, \theta>0. $$

This PSD converted to Kuş (2007) by taking α = 0.

Nasiru et al. (2019) proposed exponentiated generalized (EG) PSD by compounding exponentiated generalized distribution with power series distribution, when the components are in series i.e. using the minimum concept as given in Eq. 7. The general family is given as

$$ F(x)=1-\frac{C(\theta[1-(1-(1-G(x))^{\alpha})^{\beta}])}{C(\theta)}; \qquad \alpha, \beta, \theta>0. $$

Based on the above PSD, they proposed EG Possion class of distributions, EG binomial class of distributions, EG geometric class of distributions and EG logarithmic class of distributions.

Table 1 gives a brief on power series distributions with corresponding references, which proposed the Poisson family of distributions.

Table 1 List of research papers on Poisson power series distribution

3 Compound Method

We follow two basic principles (the minimum and the maximum) that are used in series and parallel structure. Actually, the concept of generating a new lifetime model by means of compounding was started with Adamidis and Loukas (1998). This section divided into two subsections according to condition whether the component of a system is in parallel or series, respectively.

3.1 X is Maximum

This condition occurs when the components of a system are in parallel. The method has been already discussed in Eq. 3. In this subsection, we provide a brief summary of some basic properties of proposed distributions like the nature of PDF, hazard rate, its sub-models, order relationships, limiting case, and utilized real datasets, whatever the considered model discussed.

Cancho et al. (2011) and latter Rezaei and Tahmasbi (2012) proposed Exponential Poisson (EP) distribution independently. The CDF of the EP distribution is

$$ F(x)=\frac{e^{\theta(1-e^{-\beta x}) }-1}{e^{\theta}-1} \qquad x,\theta, \beta>0. $$

(8)

The shapes of the PDF of EP distribution are either uni-model or L-shaped for a particular choice of parameters. This model has an increasing hazard rate (IHR). In real data application, Cancho et al. (2011) considered four datasets to show the utility of the model. The first dataset represents the lifetime of 23 ball bearings, shows the number of million revolutions before failure on an endurance test of deep-groove ball bearings, and was taken by Caroni (2002). The second dataset was reported by Perdona and Louzada Neto (2006), shows serum-reversal time (in days) of 143 children contaminated with HIV by vertical transmission from the University Hospital of the Ribeirão Preto, School of Medicine. The third dataset represents 213 number of successive failures of the air conditioning system of each member of a fleet of 13 Boeing 720 jet air-planes and was reported by Proschan (1963) and the fourth dataset shows the 40 lifetimes of advanced lung cancer patients, who underwent two chemotherapy treatments, but the covariates are not considered in this study and was reported by Prentice (1973). While, Rezaei and Tahmasbi (2012) also considered four datasets. The first dataset consists 18 lifetime failure data of an electronic device, reported by Wang (2000). The second dataset shows the failure times of 50 items put under a life test, reported by Aarset (1987). The third dataset shows 23 ball bearings data (Caroni, 2002), and the fourth dataset was reported by Wolford et al. (1992) and represents 20 failure times for pumps from the years 1972 to 1981.

Mahmoudi and Jafari (2012) proposed generalized exponential Poisson (GEP) distribution, by using GEPS distribution. The CDF of GEP distribution is:

$$ F(x)=\frac{e^{\theta(1-e^{-\beta x})^{\alpha} }-1}{e^{\theta}-1} \qquad x,\theta, \alpha, \beta>0. $$

The shape of the PDF of the GEP model is either uni-model or L shape, according to the choice of the parameters. The EP model is a sub-model of GEP distribution for α = 1, and the GEP distribution has an increasing hazard rate (IHR), decreasing hazard rate (DHR) and bathtub (BT) type hazard rates. In real data validation, they had considered two different real datasets. The first dataset shows 101 fatigue life of 6061-T6 aluminium coupons cut the direction of rolling and oscillated at 18 cycles per second with maximum stress per cycle 31,000 psi Birnbaum and Saunders (1969a). The second dataset consists of the 67 fatigue life (rounded to the nearest thousand cycles) of Alloy T7987 that failed before having accumulated 300 thousand cycles of testing, reported by Meeker and Escobar (2014).

Mahmoudi and Shiran (2012) and latter Mahmoudi and Sepahdar (2013) proposed exponentiated Weibull Poisson (EWP) distribution. The CDF of the distribution is

$$ F(x)=\frac{e^{\theta(1-e^{-(\beta x)^{\alpha}})^{\gamma} }-1}{e^{\theta}-1} \qquad x,\theta, \gamma, \alpha, \beta>0. $$

The techniques for proposing distribution were different, Mahmoudi and Shiran (2012) proposed EWP distribution by utilizing the concept of PSD while Mahmoudi and Sepahdar (2013) proposed by compounding technique. The EWP distribution can be expressed as an infinite linear combination of the density of exponentiated Weibull distribution, so it holds a number of mathematical properties as moments, percentiles, moment generating function, factorial moments, and others of exponentiated Weibull distribution. The shape of PDF is either uni-model or L-shaped according to the choice of parameters. The EWP is super-models of exponential-Poisson, generalized exponential-Poisson, exponentiated Rayleigh-Poisson, and Rayleigh-Poisson distributions. This has IHR, DHR, and up-side-down bathtub (UBT) type hazard rates. Mahmoudi and Shiran (2012) considered three real datasets to show the applicability of the EWP model. The first dataset shows the 67 Alloy T7987 fatigue life data Meeker and Escobar (2014). The second dataset represents 101 the fatigue life of 6061-T6 aluminium coupons data Birnbaum and Saunders (1969a). The third dataset shows 101 life of fatigue fracture of Kevlar 373/epoxy that are subject to constant pressure at the 90% stress level until all had failed and reported by Andrews and Herzberg (2012). While, Mahmoudi and Sepahdar (2013) considered two real datasets. The first dataset was taken by Smith and Naylor (1987), shows the 63 observations for strengths of 1.5 cm glass fibres, measured at the National Physical Laboratory, England, and the second dataset was 101 life of fatigue fracture of Kevlar 373/epoxy Andrews and Herzberg (2012).

Leahu et al. (2013) proposed Weibull Poisson distribution based on power series distribution. The distribution can be obtained by just putting γ = 1 in EWP Mahmoudi and Shiran (2012). The CDF of WP model is

$$ F(x)=\frac{e^{\theta(1-e^{-(\beta x)^{\alpha}}) }-1}{e^{\theta}-1} \qquad x,\theta, \alpha, \beta>0. $$

This can also be obtained from WP distribution proposed for minimum case as given in Eq. 11. They also proposed the Poisson family of distribution, the same as given in Eq. 3.

Ghorbani et al. (2014) proposed modified Weibull Poisson distribution, which is compound distribution of modified Weibull model with zero truncated Poisson distribution. The CDF of the model is

$$ F(x)=\frac{e^{\theta [1-e^{-\alpha x-\beta x^{\lambda}}]}-1}{e^{\theta}-1} \qquad x,\theta, \alpha, \beta, \lambda>0. $$

This distribution is uni-model, having IHR, DHR, and BT type hazard rates. Including modified Weibull and linear failure rate distributions, there are five sub-models of the distribution. To show the applicability of the proposed model, they had considered a real dataset of 50 failure time items reported by Aarset (1987)

Al-Zahrani and Sagor (2014) proposed Lomax Poisson (LoP) distribution. The CDF of LoP model is given as

$$ F(x)=\frac{e^{\theta[1-(1+\beta x)^{-\alpha}] }-1}{e^{\theta}-1} \qquad x,\theta, \beta, \alpha>0. $$

Although Ramos et al. (2013) proposed exponentiated Lomax Poisson distribution, it was based on the concept of the series system, i.e., the minimum method given in Eq. 5. The shape of the PDF of LoP distribution is either uni-model or L-shaped according to the choice of parameters. This distribution has DHR and UBT type hazard rates. In model validation, they had considered a real dataset of 128 remission times (in months) of bladder cancer patients and was reported by Lee and Wang (2003).

Leahu et al. (2014) proposed maximum Erlang Poisson distribution, which is based on maximum Erlang PSD. The CDF of the proposed model is

$$ F(x)=\frac{e^{\theta[1-e^{-\lambda x}{\sum}_{i=0}^{k-1}\frac{(\lambda x)^{i}}{i!}] }-1}{e^{\theta}-1}; \qquad \theta, \lambda>0. $$

The shape of the PDF of the proposed model is uni-model. They also showed that the distribution can be obtained from the limiting case of maximum Erlang binomial distribution. There is still scope for researchers to explore some of the properties of the model and its real data application.

Hassan et al. (2015) proposed complementary Burr III Poisson distribution by using the concept given in Eq. 3. They compound Burr III with zero truncated Poisson model. The CDF of the proposed model is

$$ F(x)=\frac{e^{\theta[(1+x^{-\beta})^{-\alpha}] }-1}{e^{\theta}-1} \qquad x,\theta, \beta, \alpha>0. $$

The proposed model is uni-model, having IHR and DHR types of hazard rate. Burr III distribution is a special case of this model. They performed a simulation study to show the nature of the parameters in term of mean square errors and bias.

Pararai et al. (2015a) proposed Kumaraswamy Lindley-Poisson (KLP) distribution by considering Lindley-Poisson distribution as a baseline model in Kumaraswamy family of distribution (Cordeiro and de Castro, 2011). The CDF of general form is

$$ F(x)=1-\left[ 1-\left[ \frac{e^{\theta[1-(\frac{1+\lambda+\lambda x}{1+\lambda }) e^{-\lambda x}]}-1}{e^{\theta}-1}\right]^{\alpha}\right]^{\beta} \qquad x, \theta, \beta, \alpha, \lambda>0. $$

Actually, the model proposed by Ramos et al. (2015) and Pararai et al. (2015a) are just complementary to each other and convertible to each other just by considering 𝜃 = −𝜃. The KLP distribution is right-skewed, and the PDF is either uni-model or L-shaped. This model can also be expressed as an infinite sum of a linear combination of Lindley-Poisson distribution and, in this way, holds many properties of it. Including exponentiated Lindley Poisson distribution, there are five sub-models of the proposed distribution. Also, the proposed model is very flexible and has IHR, DHR, BT and UBT type hazard rates. To show the real data application, they had considered three real datasets. The first dataset shows 119 observations on fracture toughness of Alumina (Al₂O₃) and reported by Nadarajah and Kotz (2008). The second dataset represents the 101 fatigue life of 6061-T6 aluminium coupons (Birnbaum and Saunders, 1969a). The third dataset represents the 69 tensile strength (measured in GPa), of carbon fibers tested under tension at gauge lengths of 20 mm and was analyzed by Bader and Priest (1982). The fourth dataset shows the 63 observations of 1.5 cm glass fibres strength data (Smith and Naylor, 1987).

Pararai et al. (2015b) proposed extended Lindley- Poisson distribution, which is proposed by compounding extended Lindley and zero truncated Poisson distribution. This can be obtained by using Eq. 3 and baseline as extended Lindley distribution. The CDF of the model is

$$ F(x)=\frac{e^{\theta [1-(\frac{1+\lambda+\lambda x} {1+\lambda })^{\alpha} e^{-(\lambda x)^{\beta}}]}-1}{e^{\theta}-1} \qquad x, \theta, \alpha, \beta, \lambda>0. $$

The proposed model is either uni-model or L shaped according to the choice of the parameters. This model includes Lindley Poisson, Weibull Poisson, exponential Poisson, Rayleigh Poisson, and Lindley distributions as its sub-models. The proposed model has IHR, DHR, BT, and UBT type hazard rates. To show the model applicability, they had considered a real dataset of 128 remission times of bladder cancer patients data (Lee and Wang, 2003).

Oluyede et al. (2016a) proposed four parameters Dagum-Poisson distribution by using the concept of power series distribution. This can be obtained by taking Dagum distribution as a baseline model in Eq. 3. The CDF of Dagum-Poisson distribution is

$$ F(x)=\frac{e^{\theta[1+\lambda x^{-\alpha}]^{-\beta}-1}}{e^{\theta}-1} \qquad x,\theta, \beta, \alpha, \lambda >0. $$

This model is a linear combination of Dagum densities, so it incorporates most of the properties of the Dagum model. Log-logistic Poisson, Dagum, and Burr III Poisson distributions are sub-models of the proposed model. This model is very flexible and shows IHR, DHR, UBT, and BT type hazard rates. To show the applicability of this model in real life, they had considered a dataset shows 101 life of fatigue fracture of Kevlar 373/epoxy data, reported by Andrews and Herzberg (2012).

Oluyede et al. (2016b) proposed log-logistic Weibull-Poisson distribution, by compounding log-logistic Weibull distribution with zero truncated Poisson distribution. This can be obtained by using Eq. 3 just by taking baseline model as log logistic Weibull distribution. The CDF of the model is

$$ F(x)=\frac{e^{\theta [1-(1+(\frac{x}{s})^{c})^{-1} e^{-\lambda x^{\beta}}]}-1}{e^{\theta}-1} \qquad x, \theta, \beta, s, c, \lambda>0. $$

The shape of the distribution is either uni-model or L-shaped for a particular choice of parameters. Including Weibull and exponential distribution, there are ten sub-models can be obtained form this model. The proposed distribution have five parameter model, having various type of hazard rates like IHR, DHR, UBT, and BT type hazard rates. In the real data section, they had considered two real datasets. The first dataset shows 101 observations of the tensile fatigue characteristics of a polyester/viscose yarn, reported by Picciotto and Hersh (1972). The second dataset refers to 101 life of fatigue fracture of Kevlar 373/epoxy data, reported by Andrews and Herzberg (2012).

Tahir and Cordeiro (2016) proposed exponentiated Kumaraswamy Poisson family of distribution, McDonald and beta generated Poisson family of distributions. The CDF of general form of exponentiated Kumaraswamy PFD is

$$ F(x)=\frac{e^{\theta K^{\alpha}(x)}-1}{e^{\theta}-1} \qquad x, \alpha, \theta>0. $$

(9)

where K(x) is Kumaraswamy family of distribution. In the same fashion, McDonald PFD and beta PFD are defined by just considering baseline CDF as McDonald family and beta family of distributions in Eq. 3.

There are some scope for researchers to explore the statistical properties and real data applications for these distributions.

Pararai et al. (2017) proposed exponentiated power Lindley Poisson distribution, which is obtained by compounding exponentiated power Lindley distribution with zero truncated Poisson distribution. This can be obtained by just considering exponentiated power Lindley distribution as a baseline model in Eq. 3.

The CDF of the proposed model is

$$ F(x)=\frac{e^{\theta [1-(\frac{1+\lambda+\lambda x^{\alpha}}{1+\lambda }) e^{-(\lambda x^{\alpha})}]^{\beta}}-1}{e^{\theta}-1} \qquad x, \theta, \alpha, \beta, \lambda>0. $$

This model can be expressed as an infinite linear combination of the density of exponentiated power Lindley distribution, so it holds various mathematical properties as moments, percentiles, moment generating function, factorial moments, and others of exponentiated power Lindley distribution. This is a uni-model distribution and also very flexible having IHR, DHR, UBT, and BT type hazard rates. Including power Lindley–Poisson and exponentiated Lindley–Poisson distributions, six sub-models, can be obtained from the proposed model. This model also holds likelihood ratio ordering property and, in this way, also holds a stochastic order relationship for keeping all the parameters being equal except 𝜃. To show the model applicability, they had considered two different datasets. The first dataset shows the strength of 1.5 cm glass fibres (Smith and Naylor, 1987), and the second dataset shows 100 observations on breaking stress of carbon fibres given by Nichols and Padgett (2006).

Yousof et al. (2018) proposed transmuted exponential Poisson distribution, which is transmuted form of Cancho et al. (2011). The CDF of the distribution is

$$ F(x)=\frac{[e^{\theta(1-e^{-\beta x})}-1]\left[1+\lambda\frac{e^{\theta}-e^{\theta(1-e^{-\beta x})}}{e^{\theta}-1} \right] }{e^{\theta}-1} \qquad x,\theta, \beta>0, |\lambda|\le 1. $$

They also proposed a new generalized class of transmuted Poisson family of distribution by using the concept of T-X family of distribution in transmuted Poisson family of distribution. The CDF of general form of new class of distribution is

$$ F(x)={\int}_{0}^{-\log[1-G(x)]}\frac{\theta e^{-t}e^{\theta(1-e^{-t})}}{e^{\theta}-1}\left[1-\lambda+2\lambda\frac{e^{\theta}-e^{\theta(1-e^{-t})}}{e^{\theta}-1} \right]dt \quad x,\theta>0, |\lambda|\le 1. $$

Based on the above transformation, they also proposed generalized transmuted Poisson (GTP) Weibull and GTP Lindley distributions. Actually, the model proposed by Alizadeh et al. (2018) and this model is complimentary to each other and one can be obtained by another just by putting 𝜃 = −𝜃 in each other. In the similar way, the proposed GTP Weibull and GTP Lindley models can obtained by using the Eqs. 13 and 14. The proposed GTP Weibull and GTP Lindley distributions are either uni-model or L shaped PDF and the hazard rate is different. The hazard rate is IHR and DHR for GTP Weibull while for GTP Lindley it is IHR, DHR and UBT type nature. In real data application they had considered two real dataset. The first represents relief times of 20 patients receiving an analgesic given by Gross and Clark (1975) and second dataset shows 346 nicotine measurements (in 1998), made from several brands of cigarettes, collected by the Federal Trade Commission which is an independent agency of the US government and can be found from http://pw1.netfom.com/rda vis2/smoke.html.

Table 2 gives a brief on the Poisson family of distributions based on the maximum concept with corresponding references.

Table 2 List of research papers on Poisson family of distribution (max)

3.2 X is Minimum

This condition occurs when the components of a system are in series. The method is already discussed in Eq. 5. In this subsection, we have also provided a brief summary of some basic properties of proposed distributions like its nature of PDF, hazard rate, sub-models, order relationships, limiting case, and utilized real dataset, whatever they proposed.

Kuş (2007) has also proposed exponential-Poisson (EP) distribution by second transformation defined in Eq. 5. The CDF of the model is

$$ F(x)=\frac{1-e^{-\theta(1-e^{-\lambda x})}}{1-e^{-\theta}} \qquad x, \lambda, \theta >0. $$

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The shape of the PDF of the EP model is the same as the shape of the PDF of exponential distribution i.e., L shaped. The difference between these two EP models is that the former is IHR, while the latter is DHR. The beauty of its hazard rate is that the initial and long term hazards function both are finite i.e., the hazard rate is bounded on both sides. In real data analysis, they had considered two datasets. The first dataset represents 213 number of successive failures of the air conditioning system (Proschan, 1963), and the second dataset represents 109 observations on the period between successive coal-mining disasters and is reported by Cox and Lewis (1978).

Barreto-Souza and Cribari-Neto (2009) generalized EP distribution by rising one additional parameter and proposed exponentiated exponential Poission (EEP) distribution. The CDF of EEP distribution is

$$ F(x)=\left[ \frac{1-e^{-\theta(1-e^{-\beta x}) }}{1-e^{-\theta}}\right]^{\alpha} \qquad x, \alpha, \beta, \theta>0. $$

The EEP has increasing, decreasing and upside down bathtub hazard rate distribution. The EP distribution can be obtained by just putting the parameter shape parameter to the unit, and EEP distribution can be written as a linear combination of the EP model. The nature of the PDF is either uni-model or L shape according to the choice of parameter. The shape of the hazard rate is IHR, DHR, and UBT types. To show the applicability of the model, they had considered two real datasets. The first dataset shows 30 successive values of March precipitation (in inches) in Minneapolis/St Paul, reported by Hinkley (1977). The second dataset shows 31 the prices of children’s wooden toys on sale in a Suffolk craft shop in April 1991 and was reported by The Open University (1963).

Preda et al. (2011) proposed modified exponential Poisson distribution. Modification in the sense that, they had considered Marshal-Olkin exponential distribution (Marshall and Olkin, 1997) and compound this with Poisson distribution. The CDF of the proposed model is

$$ F(x)=\frac{1-e^{-\theta +\frac{\alpha \theta e^{-\beta x}}{1-\bar{\alpha} e^{-\beta x}}}}{1-e^{-\theta}} \qquad x,\theta, \alpha, \beta>0 $$

where $\bar {\alpha }=1-\alpha$. The shape of the model is either uni-model or L-shaped according to the parameter choice, and the EP model is a sub-model of the proposed model. The shape of the hazard rate is IHR, DHR, and UBT type. There is still some scope for researchers to establish their real data applications in various fields of life.

Hemmati et al. (2011) and Bereta et al. (2011) and latter Lu and Shi (2012) proposed independently, Weibull Poisson (WP) model, which is generalization of Kuş (2007) proposed model. All have proposed same model with similar concept. The CDF of WP model is

$$ F(x)=\frac{1-e^{-\theta [1-e^{-(\alpha x)^{\gamma}}]}}{1-e^{-\theta}} \qquad x, \theta, \alpha, \gamma>0. $$

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The WP model can be represented as an infinite sum of a linear combination of Weibull distribution and, in this way, holds many properties of it. The hazard rate of the WP model is IHR, DHR, and modified UBT types. The modified UBT models are often found in mixture distribution such as mixtures of Weibull and Gamma models. The EP, Rayleigh–Poisson, Weibull models are the sub-models of the WP model. The shape of the PDF is uni-model, and L shaped according to the choice of the parameters. The WP model can be obtained as a limiting case of Weibull binomial distribution. In model validation, Hemmati et al. (2011) considered a real data set of 100 observations on breaking stress of carbon fibres given by Nichols and Padgett (2006). Bereta et al. (2011) considered a real dataset represents 213 number of successive failures of the air conditioning system (Proschan, 1963). Lu and Shi (2012) used three datasets to show the applicability of the model. The first considered dataset is same as considered by Bereta et al. (2011). The second dataset shows 30 successive values of March precipitation (Hinkley, 1977), and the third dataset represents 46 active repair times (in hours) for an airborne communication transceiver taken by Von Alven (1964).

Morais and Barreto-Souza (2011) proposed a compound family of Weibull and power series distributions. By using the WPS class of distribution, they also proposed Weibull Poisson (WP) distribution the same as given in Eq. 11. Actually, the concept of generating a new lifetime model by means of rising power by Gupta et al. (1998). Weibull distribution is a special limiting case of WPS distribution. The PDF of WPS distribution is L shaped when the shape parameter γ ≤ 1 in Eq. 11 and has at least one mode if γ > 1. In real data applications, they had considered two real datasets. The first dataset shows the 63 observations 1.5 cm glass fibres strength data, taken by Smith and Naylor (1987), and the second dataset represents 101 fatigue life of 6061-T6 aluminium coupons, reported by Birnbaum and Saunders (1969a).

Alkarni and Oraby (2012) also proposed the Poisson generated family of distribution by using the concept given in Eq. 5. The transformation proposed by Kumar et al. (2015) seems to be complementary form of this method. They also discussed the form of Poisson exponential, Weibull Poisson, Rayleigh Poisson and Pareto Poisson models. The CDF of Pareto Poisson model is

$$ F(x)=\frac{1-e^{-\theta [1-(1+x)^{-\beta}]}}{1-e^{-\theta}} \qquad x\geq 1, \theta, \beta>0. $$

There is a scope for researchers to explore the statistical properties and real data application for the proposed distribution.

Cordeiro et al. (2012) proposed exponential Conway Maxwell-Poisson distribution, by using the similar concept obtained by Kuş (2007). Initially, Rodrigues et al. (2009) proposed Conway Maxwell (COM)-Poisson cure rate survival model, which is an extension of Poisson model and incorporates some of the well-known cure rate models. This model incorporates exponential geometric and exponential Poisson distributions as its sub model. The CDF of the proposed model is

$$ F(x)=1-\frac{Z(\theta e^{-\beta x}, \nu)}{Z(\theta, \nu)-1}, \quad \nu\ge0 $$

where $Z(\theta , \nu )={\sum }_{j=0}^{\infty }\frac {\theta ^{j}}{(j!)^{\nu }}$ is a generalization of several well-known infinite sums. Various distribution can be obtained on the basis of value of ν as ν = 1 it represent Poisson model, ν > 1 represent under-dispersion and ν < 1 represent over-dispersion of Poisson model; for $\nu =\infty$ it represent Bernoulli distribution with parameter (1 + 𝜃)^− 1. And for ν = 0,𝜃 < 1 it convert to geometric distribution having parameter 1 − 𝜃. The COM Poisson distribution converted to Kuş (2007) for ν = 1 and Adamidis and Loukas (1998) for ν = 0. In real data application, they had considered a dataset representing 427 cutaneous melanoma (a type of malignant cancer) patients data for the evaluation of postoperative treatment performance with a high dose of a certain drug (interferon alfa-2b) in order to prevent recurrence from 1991 to 1995 and was taken by Ibrahim et al. (2014).

Leahu et al. (2013) proposed gamma Poisson distribution based on PSD. The CDF of the model can be obtained by considering baseline as gamma distribution in Eq. 5. The CDF of the model is given as

$$ F(x)=\frac{1-e^{-\theta [{{\int}_{0}^{x}}\frac{e^{-y}y^{\lambda-1}}{\Gamma\lambda}dy]}}{1-e^{-\theta}} \qquad x, \theta, \lambda>0. $$

They also proposed the Poisson family of distribution, the same as given in Eq. 5.

Percontini et al. (2013) proposed beta Weibull Poisson distribution by compounding beta and Weibull Poisson distribution. They have considered Weibull Poisson distribution (Lu and Shi, 2012) as baseline model in beta generated family of distribution having CDF as

$$ F(x)=\frac{1}{B(a, b)}{\int}_{0}^{G(x)}y^{a-1}(1-y)^{b-1}dy \quad a, b>0 $$

where B(a,b) is beta function. Including beta exponential Poisson and exponentiated Weibull Poisson distributions, there are eighteen sub-models of the proposed model. The nature of the PDF is either uni-model or L-shaped according to the choice of the parameters, and the shape of the hazard rates are IHR, DHR, and UBT types. In real data application, they had considered a dataset represents 46 active repair times for an airborne (Von Alven, 1964).

Silva et al. (2013) proposed Pareto Poisson distribution by using extended Weibull power series distribution. The CDF of model is

$$ F(x)=\frac{1-e^{-\theta [1-(\alpha/ x)^{\beta}]}}{1-e^{-\theta}} \qquad x\geq\alpha, \theta, \beta>0. $$

The Pareto distribution is a sub-model of the proposed model. The Pareto Poisson model has DHR and BT type hazard rates. To show the applicability of the model, they had considered a real dataset of 128 soil fertility influence and the characterization of the biologic fixation of N₂ for the Dimorphandra wilsonii rizz growth and taken by Fonseca and Franca (2007).

Nadarajah et al. (2013) proposed three parameter geometric exponential Poisson distribution, which is compound distribution of geometric exponential model with zero truncated Poisson distribution. The CDF is given as

$$ F(x)=\frac{e^{-\theta e^{-\alpha x}}-e^{-\theta}}{1-e^{-\theta}-\lambda[1-e^{-\theta e^{-\alpha x}}]} \qquad x, \alpha, \theta >0, 0<\lambda<1. $$

The exponential geometric and exponential Poisson (Kuş, 2007) models are sub-model of the proposed model. The lower tail tends to a constant value while the upper tail of the PDF decreases exponentially. The hazard rate of the model is IHR, DHR, and UBT types. They had considered a real dataset of 690 adult numbers of Tribolium confusum (Eugene et al. 2002) to show superior performance with some other lifetime models.

Ramos et al. (2013) exponentiated Lomax Poisson distribution, which is obtained by compounding exponentiated Lomax distribution (Abdul-Moniem and Abdel-Hameed, 2012) with zero truncated Poisson distribution by using the method given in Eq. 5. The CDF of the model is

$$ F(x)=\frac{1-e^{-\theta [1-(1+\alpha x)^{-\beta}]^{\gamma}}}{1-e^{-\theta}} \qquad x, \alpha, \theta, \beta, \gamma>0. $$

Including exponentiated Pareto and exponentiated Lomax distributions, there are four sub-models of the exponentiated Lomax Poisson distribution. The graphical analysis shows that the proposed model shows either uni-model of L shaped PDF, and the hazard rate exhibit DHR and UBT types. In real data application, they had considered a dataset reported by Murthy et al. (2004), representing 88 failure times for a particular wind-shield model.

Barreto-Souza and Simas (2013) also proposed the distribution obtained in Eq. 5 by considering truncated exponential distribution between interval [0, 1] and then generalized it for general family of distribution. The CDF of general family of distribution is

$$ F(x)=\frac{1-e^{-\theta G(x)}}{1-e^{-\theta}} \qquad x, \theta>0. $$

They consider Weibull distribution as baseline model and proposed a new distribution which possesses modified bathtub hazard rate. The functional form of exponential Weibull distribution is same as Weibull Poisson distribution proposed by Hemmati et al. (2011), Bereta et al. (2011), and Lu and Shi (2012) given in Eq. 11. So the distributional properties are the same as for the WP model. In real data application, they had considered a real of 101 fatigue life of 6061-T6 aluminium coupons data, reported by Birnbaum and Saunders (1969a).

Nadarajah et al. (2014) proposed truncated-exponential skew-symmetric (TESS) distribution, whose general form of the CDF is

$$ F(x)=\frac{1-e^{-\theta G(x)}}{1-e^{-\theta}} \qquad x,\theta>0. $$

The form and methodology of proposing TESS is same as Barreto-Souza and Simas (2013). They proposed TESS normal, TESS t, and TESS Cauchy distributions by considering normal, t, and Cauchy distributions as a baseline model. However, the weakness of TESS distributions that their skewness and kurtosis measures are not monotonic with respect to parameter |𝜃|.

Ristić and Nadarajah (2014) proposed generalized-exponential Poisson (GEP) distribution by using the minimum concept. The CDF of the distribution is

$$ F(x)=\frac{1-e^{-\theta(1-e^{-\lambda x})^{\alpha}}}{1-e^{-\theta}} \qquad x, \lambda, \alpha, \theta >0 $$

which is generalization of Kuş (2007) model and for particular choice of parameter α = 1 it reduces to exponential Poisson model. The lower tail behaviour of the density of the model exhibits like the gamma distribution, and the upper tail exhibit the exponential distribution. The hazard rate function is initially shown polynomial nature and become constant as $x\rightarrow \infty .$ The proposed model has IHR, DHR, and UBT type hazard rates. In real data application, they had considered a real dataset of cairo available in R software having 306 observations.

Gupta et al. (2014) proposed exponential generalized Poisson distribution, which is compound distribution of exponential model with zero truncated generalized Poisson distribution. This model shows decreasing hazard rate. The CDF of the proposed model is

$$ F(x)=\frac{1-e^{-\frac{\theta }{\alpha}[W(g(x))+\alpha] }}{1-e^{-\theta}} \qquad x, \alpha, \theta, \beta>0 $$

where g(x) = −αe^−α−βx and W(⋅) is Lambert W function, which is defined as W(x)e^W(x) = x. This model is a mixture of the exponential distribution. The nature of the PDF of the model is L shaped. In real data application, they had considered two datasets. The first dataset shows 24 observations on the time intervals between successive earthquakes in the 20th century in the North Anatolia fault zone and was analyzed by Kuş (2007). The second dataset shows 213 observations of a number of successive failures of the air conditioning system (Proschan, 1963).

Gui et al. (2014) proposed Lindley Poisson (LP) distribution by considering Lindley model as baseline in Eq. 5.

$$ F(x)=\frac{1-e^{-\theta [1-(\frac{1+\alpha+\alpha x}{1+\alpha })e^{-\alpha x}]}}{1-e^{-\theta}} \qquad x, \theta, \alpha>0. $$

The PDF plot of the Lindley–Poisson distribution is either uni-model or decreasing, according to the choice of parameters. Also, the PDF is right-skewed and has a constant mass at the initial point. Lindley distribution is a sub-model of it. The proposed model also holds a likelihood ratio order relationship and, in this way, also holds a stochastic order relationship. This model has IHR and UBT type hazard rates. In real data applications, they had considered two real datasets. The first dataset shows 24 observations on the time intervals between successive earthquakes data (Kuş, 2007), and the second dataset shows the survival times (in days) of 72 guinea pigs infected with virulent tubercle bacilli reported by Bjerkedal (1960).

Hashimoto et al. (2014) proposed Birnbaum-Saunders Poisson distribution, which is compound distribution of Birnbaum-Saunders distribution (Birnbaum and Saunders, 1969b) with zero truncated Poisson model utilizing the minimum concept given in Eq. 5. The CDF of the proposed model is

$$ F(x)=\frac{1-e^{-\theta{\Phi}[(1/\alpha)(\sqrt{x/\lambda}-\sqrt{\lambda/x})] }}{1-e^{-\theta}} \qquad x,\theta, \alpha, \lambda>0 $$

where Φ(⋅) is standard normal cumulative distribution function. The nature of the PDF of the proposed model is uni-model. This model is a cure rate model that shows a UBT shaped hazard rate. In real data application, they had considered a dataset of 45 female breast cancer patients with negative axillary lymph nodes and a minimum 10-year follow-up were selected from the Ohio State University Hospital Cancer Registry and reported by Sedmak et al. (1989).

Leahu et al. (2014) proposed a minimum Erlang Poisson distribution, which is based on minimum Erlang PSD. The CDF of the proposed model is given as

$$ F(x)=\frac{1-e^{-\theta[1-e^{-\lambda x}{\sum}_{i=0}^{k-1}\frac{(\lambda x)^{i}}{i!}] }}{1-e^{-\theta}}; \qquad \theta, \lambda>0. $$

The shape of the PDF of the proposed model is uni-model. They also showed that the distribution can be obtained from the limiting case of maximum Erlang binomial distribution. There is still a scope to explore some of the properties of the model and its real data application for researchers.

Cordeiro et al. (2015) Poisson generalized linear failure rate model, which can be obtained by compounding generalized linear failure rate (GLFR) model with Poisson distribution. The generalized linear failure rate model is Lehman type I (alpha power) generalization of linear failure rate model. The CDF of the proposed model is

$$ F(x)=\frac{1-e^{-\theta\left[ 1-e^{-\beta x-\frac{\gamma^{2}}{2}x^{2}} \right]^{\alpha} }}{1-e^{-\theta}} \qquad x,\theta, \gamma, \alpha, \beta >0. $$

GLRF Poisson model is a supermodel of the GLRF model and can be expressed as an infinite sum of a linear combination of the GLRF model. It is a very flexible model having uni-model and L-shaped density and IHR, UBT, and BT type hazard rates. In real data application, they had considered a dataset representing 427 cutaneous melanoma cancer patients data, taken by Ibrahim et al. (2014).

Delgarm and Zadkarami (2015) proposed modified Weibull Poisson distribution. This model is compounding of modified Weibull (Lai et al. 2003) with zero truncated Poisson distribution by means of method given in Eq. 5. The CDF of the proposed model is

$$ F(x)=\frac{1-e^{-\theta [1-e^{-\alpha x^{\beta} e^{\lambda x}}]}}{1-e^{-\theta}} \qquad x,\theta, \alpha, \beta, \lambda>0. $$

This model is different from Ghorbani et al. (2014) in the sense that it uses Lai et al. (2003) model with method given in Eq. 5 while Ghorbani et al. (2014) utilises Sarhan and Zaindin (2009) model with the method given in Eq. 3. The PDF of the modified Weibull Poisson model is either uni-model or L-shaped according to the parameter choices. The WP and modified Weibull models are the sub-model of the proposed model. This distribution has IHR, DHR, BT, and UBT type hazard rates. To show the applicability of the model, they had considered two real datasets. The first dataset shows 50 failure time items, reported by Aarset (1987), while the second dataset shows the failure (in minutes) of 15 electronic components in an accelerated life test and is reported by Lawless (2011).

Silva and Cordeiro (2015) proposed Burr XII (Zimmer et al. 1998)- Poisson (BXIIP) distribution by using the concept of zero truncated power series distributions utilized by previously Morais and Barreto-Souza (2011) and Silva et al. (2013). The CDF of BXIIP model is

$$ F(x)=\frac{1-e^{-\theta\left[ 1-(1+x^{c})^{-k} \right] }}{1-e^{-\theta}} \qquad x,\theta, k, c >0. $$

BXIIP is generalization of Weibull-Poisson (WP) distribution (when 1/k tends to zero), exponential-Poisson (EP) when 1/k tends to zero with c = 1 and Burr XII distribution when 𝜃 tends to zero. The PDF of the BXIIP model is either uni-model or L-shaped according to the parameter choices, and the hazard rates of are DHR and UBT type. To show the applicability of the model, they had considered two real datasets. The first dataset represents 1519 observations of budget share for fuel expenditure of British households (1980—1982) was taken by Blundell et al. (1998). The second dataset consists of the failure times of 20 mechanical components and taken by Murthy et al. (2004).

Silva et al. (2015) proposed five parameter exponentiated Burr XII Poisson distribution (Exp-BXIIP). The CDF of the model is

$$ F(x)=\frac{1-e^{-\theta\left[ 1-\left[1+(\frac{x}{s})^{c} \right]^{-k} \right]^{\alpha} }}{1-e^{-\theta}} \qquad x,\theta, \alpha, k, c, s >0. $$

The PDF of the BXIIP model is either uni-model or L-shaped according to the parameter choices. It can be represented as a linear combination of an infinite sum of exponentiated Burr XII distributions, so it holds some of its properties. The sub-models of Exp-BXIIP are logistic, log-logistic, Weibull, Burr XII, and exponentiated Burr XII, BXIIP distributions. When 𝜃 tends to zero, we get the base distribution. This is a very flexible model having various types of hazard rates like IHR, DHR, BT, and UBT type hazard rates. To show the applicability of the model, they had considered a real dataset of the failure times of 20 mechanical components and was taken by Murthy et al. (2004).

Gomes et al. (2015) also proposed exponentiated generalized Poisson distribution independently, which is form of Eq. 5 with exponentiated family of distribution. The CDF of the general form of distribution is

$$ F(x)=\frac{1-e^{-\theta G^{\alpha}(x) }}{1-e^{-\theta}} \qquad x, \alpha, \theta>0. $$

They also proposed exponentiated Burr XII Poisson (Exp-BXIIP) and exponentiated Weibull Poisson (Exp-WP) distribution by considering baseline as Burr XII and Weibull model respectively. The Exp-BXIIP model is same as Silva et al. (2015) and the CDF of Exp-WP model is

$$ F(x)=\frac{1-e^{-\theta [1-e^{-(\alpha x)^{\beta}}]^{\gamma}}}{1-e^{-\theta}} \qquad x,\theta, \alpha, \beta, \gamma>0. $$

The proposed general family of distribution can be expressed as an infinite sum of a linear combination of the exponentiated family of distribution and, in this way, holds many properties of it. For more details about Exp-BXIIP model see Silva et al. (2015). They used 104 life-history attributes for the mountain brush-tail possum (Trichosurus caninus) in mountain ash (Eucalyptus regnans) forest at Cambarville in the Central Highlands of Victoria, south-eastern Australia. This data frame consists of nine morphometric measurements, and they used measurements of skull width. This is available in R library DAAG.

Ramos et al. (2015) proposed Kumaraswamy-generated Poisson (KGP) family of distribution. The CDF of KGP family of distribution is

$$ F(x)=\frac{1-e^{-\theta[1-(1-G^{\alpha}(x))^{\beta}]}}{1-e^{-\theta}} \qquad x, \theta, \beta, \alpha>0. $$

They considered Weibull, gamma, and beta distributions as baseline model and proposed KGP Weibull, KGP gamma, and KGP beta distributions. The PDF of KGP Weibull and KGP gamma models are either uni-model or L-shaped, while for KGP beta distribution, it has J shaped nature also. The hazard rate of KGP Weibull and KGP gamma models are IHR, DHR, and UBT types. In real data application, they had considered one dataset and KGP Weibull model only. The dataset represent 128 remission times of bladder cancer patients data (Lee and Wang, 2003).

Gitifar et al. (2016) proposed linear failure rate Poisson (LFRP) distribution by method of compounding LFR distribution (Bain, 1974) compounding with truncated Poisson distribution. The CDF of the LFRP is

$$ F(x)=\frac{1-e^{-\theta\left[ 1-e^{-\beta x-\frac{\gamma^{2}}{2}x^{2}} \right] }}{1-e^{-\theta}} \qquad x,\theta, \gamma, \beta >0. $$

Latter, Mahmoudi and Jafari (2017) also proposed the same by utilizing the concept of PSD, and the generalized form was proposed by Cordeiro et al. (2015). The PDF of the LFRP model is either uni-model or L-shaped according to the parameter choices, and the hazard rates of are IHR, DHR, BT, and modified BT type. The LFR model is a sub-model of the proposed model. They had considered two real datasets to show the applicability of the model. The first dataset represented the lifetime (in days) of 40 patients suffering from leukemia from one of the Ministry of Health Hospitals in Saudi Arabia and was taken by Abouammoh et al. (1994). The Second dataset shows the 63 observations of 1.5 cm glass fibres’ strengths data, taken by Smith and Naylor (1987).

Tahir et al. (2016) proposed Poisson-X family of distribution which was based on the compounding zero truncated Poisson distribution with T-X family of distribution (Alzaatreh et al. 2013). The CDF of the general form of the class of distribution is

$$ F(x)=\frac{1-e^{- G^{\alpha}(x)}}{1-e^{-1}} \qquad x, \alpha>0. $$

They also proposed power Cauchy Poisson (PCP) distribution, based on the above method. The CDF of PCP distribution is

$$ F(x)=\frac{1-e^{- [2\pi^{-1} tan^{-1}(\frac{x}{\lambda})^{\beta}]^{\alpha}}}{1-e^{-1}} \qquad x, \alpha, \lambda, \beta>0. $$

The PCP distribution is very flexible and exhibits left-skewed, right-skewed, reversed-J, and uni-model PDF, and the hazard rates are IHR, DHR, UBT, and BT types. In real data application, they had considered three datasets. The first dataset shows annual food discharge rates for the 39 years (1935-1973) at Floyd River located in James, Iowa, USA, and was reported (Mudholkar and Hutson, 1996). The second dataset shows the 63 observations of 1.5 cm glass fibers’ strengths data, was taken by Smith and Naylor (1987), and the third dataset shows 101 life of fatigue fracture of Kevlar 373/epoxy data, reported by Andrews and Herzberg (2012).

Motivated by Gomes et al. (2015), Tahir and Cordeiro (2016) proposed exponentiated Kumaraswamy Poisson family of distribution, McDonald and beta generated Poisson family of distributions. The general form of exponentiated Kumaraswamy PFD is

$$ F(x)=\frac{1-e^{-\theta K^{\alpha}(x)}}{1-e^{-\theta}} \qquad x, \alpha, \theta>0, $$

where K(x) is Kumaraswamy family of distribution. For particular choice of shape parameter α = 1 it reduced to Ramos et al. (2015). The general form of McDonald PFD is

$$ F(x)=\frac{1-e^{-\theta M(x)}}{1-e^{-\theta}} \qquad x, \theta>0, $$

where M(x) is McDonald family of distribution. And similarly beta PFD is proposed by just replacing baseline CDF by beta family of distribution in Eq. 5. There is a scope for researchers to explore the statistical properties and real data applications for these distributions.

Vigas et al. (2017) proposed Weibull Poisson regression model based on WP model given in Eq. 11. The CDF of the proposed model is

$$ F(x)=\frac{1-exp(-\theta [1-e^{-e^{(\frac{y-\boldsymbol{X}\beta}{\sigma})}}]) }{1-e^{-\theta}} \qquad -\infty< y, \mu< \infty,\theta, \sigma, \beta>0. $$

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This can be defined from Eq. 11 by just putting α = e^−μ; γ = 1/σ; explanatory variable vector is denoted by X is a known model matrix, which is related to response $y=\log x$ and μ = Xβ. They considered a simulation study to show the performance of the model with the Weibull and exponential-Poisson regression models in terms of mean square errors. Also, they had considered sensitivity analysis from case deletion to conduct a robustness study to detect in influential or extreme observations. In real data analysis, they had considered a real dataset, reported by Crowley and Hu (1977) of 103 survival time (in days) of heart transplant patients data (1967- 1973) who were admitted into the Stanford University.

Aryal and Yousof (2017) proposed exponentiated generated Poisson (EGP) family of distribution. The CDF of EGP family of distribution is

$$ F(x)=\frac{1-e^{-\theta[(1-(1-G(x))^{\alpha})^{\beta}]}}{1-e^{-\theta}} \qquad x, \theta, \beta, \alpha>0. $$

They considered Weibull and Pareto distributions as a baseline model and proposed EGP Weibull and EGP Pareto distributions. The CDF of EGP Weibull model is

$$ F(x)=\frac{1-e^{-\theta[(1-(e^{-ax^{b}})^{\alpha})^{\beta}]}}{1-e^{-\theta}} \qquad x, \theta, \beta, \alpha, a, b>0. $$

And the CDF of EGP Pareto model is

$$ F(x)=\frac{1-e^{-\theta[(1-(\frac{b}{x})^{a \alpha})^{\beta}]}}{1-e^{-\theta}} \qquad x\le b, \theta, \beta, \alpha, a>0. $$

The EGP family of distribution is generalization of Kumaraswamy-generated Poisson (KGP) family of distribution proposed by Ramos et al. (2015). The PDF of both distributions are more flexible to its baseline and exhibit either uni-model or L shaped nature, according to the choice of the parameters. There is still some scope to study its properties, including the hazard rate. In real data application, they had considered two datasets. The first dataset represent 128 remission times of bladder cancer patients data (Lee and Wang, 2003). The second dataset shows the 72 exceedances of flood peaks (in m³/s) of the Wheaton River near Carcross in Yukon Territory, Canada (1958—1984) reported by Choulakian and Stephens (2001).

Alizadeh et al. (2018) proposed generalized transmuted Poisson (GTP) class of distribution. The general form of proposed model is given as

$$ F(x)=\frac{e^{\theta}-e^{\theta \bar{G(x)}}}{e^{\theta}-1}\left[ 1+\lambda\frac{e^{\theta \bar{G(x)}}-1}{e^{\theta}-1}\right]\qquad \theta>0, |\lambda|\leq1 $$

where $\bar {G(x)}=1-G(x).$ Based on the above technique, they also proposed GTP Weibull and GTP Lindley distributions. The CDF of GTP Weibull distribution is

$$ F(x)=\frac{e^{\theta}-e^{\theta e^{-x^{\beta}} }}{e^{\theta}-1}\left[ 1+\lambda\frac{e^{\theta e^{-x^{\beta}}}-1}{e^{\theta}-1}\right]\qquad \theta, \beta>0, |\lambda|\leq1. $$

(13)

And the CDF of the GTP Lindley distribution is

$$ F(x)=\frac{e^{\theta}-e^{\theta (\frac{1+\alpha+\alpha x}{1+\alpha })e^{-\alpha x} }}{e^{\theta}-1}\left[ 1+\lambda\frac{e^{\theta (\frac{1+\alpha+\alpha x}{1+\alpha })e^{-\alpha x}}-1}{e^{\theta}-1}\right]\qquad \theta, \alpha>0, |\lambda|\leq1. $$

(14)

The GTP family of distribution can be expressed as infinite sum of linear combination of exponentiated family of distribution and in this way holds many properties of it. As $x\rightarrow \infty$, the PDF of GTP family of distribution converges to baseline distribution with some weight and the hazard converges to the hazard rate of the baseline model. The nature of PDF of both the distribution shows either uni-model or L shaped, according to the parameters choice. The hazard rate of the GTP Weibull distribution has IHR, DHR and UBT type, while the GTP Lindley distribution has IHR, DHR and BT type hazard rates. In real data application, they had considered two real datasets. The first dataset shows 40 active repair times (in hours) taken by Jorgensen (2012), and the second dataset shows the 128 remission times of bladder cancer patients data (Lee and Wang, 2003).

Nasiru et al. (2019) proposed exponentiated generalized power series distribution (EGPSD) and based on it, they proposed exponentiated generalized Poisson inverse exponential (EGPIE) distribution. The CDF of proposed model is

$$ F(x)=\frac{1-e^{-\theta[(1-(1-e^{-\frac{\lambda}{x}})^{\alpha})^{\beta}]}}{1-e^{-\theta}}; \qquad \alpha, \beta, \theta, \lambda>0. $$

The exponentiated generalized class of distribution and exponentiated PSD are the limiting cases of EGPSD. This family of distributions is also holding the property of the likelihood ratio order relationship. The EGPIE distribution has right-skewed, uni-model shaped PDF, and the hazard rate function exhibit UBT type nature. In real data applications, they had considered two real datasets. The first dataset shows 101 life of fatigue fracture of Kevlar 373/epoxy data, reported by Andrews and Herzberg (2012). The second dataset shows 59 monthly actual tax revenue in Egypt (January 2006 to November 2010) in 1000 million Egyptian pounds, reported by Mead (2014).

Table 3, gives a brief on Poisson family of distributions based on minimum concept with their references.

Table 3 List of research papers on Poisson family of distribution (min)

Table 4, represent the list of datasets utilized in corresponding models with reference and sample sizes. The column Data Description shows the brief details of datasets; Reference, shows the corresponding references of the datasets. The last column Model, shows the corresponding model that utilizes the datasets; the $\max \limits$ shows that the model is defined for maximum criterion as given in Eq. 3 and $\min \limits$ shows that the model is defined for minimum criterion as given in Eq. 5. This table shows that there are 40 datasets used by 54 authors in 62 models.

Table 4 Table for dataset modelled by geometric family of distributions

4 Transformations Based on PGF of Distribution

There are some transformation methods are available in literature which are anyhow related to Poisson family of distribution. But the noted point is that, non of the methods shows its relation with this family. Here, we are providing the complete list of these transformation methods.

4.1 Alpha Logarithmic Transformation

Motivated by Marshall and Olkin (1997) & Pappas and Adamidis (2012) proposed a new transformation given as

$$ S(x)=\frac{\log(1+(\alpha-1)S_{0}(x))}{\log(\alpha)} \qquad x, \alpha>0. $$

where S(x) is new survival function and S₀(x) is baseline survival function. This transformation can be obtained by Poisson family of distribution. By reverting Eq. 3, we get,

$$ \begin{array}{lll} 1+ (e^{\theta}-1) F(x)&=e^{\theta G(x)} \qquad x, \theta>0\\ G(x)&=\frac{ \log(1+(e^{\theta}-1) F(x))}{\theta}\\ \text{by putting } \theta&=\log(\alpha)\\ G(x)&=\frac{ \log(1+ (\alpha-1) F(x))}{\log(\alpha)} \end{array} $$

(15)

Now, by applying Lehmann type II (i.e. Survival concept), AL transformation can be obtained. Dey et al. (2017b) proposed AL generalized exponential distribution and studied its statistical properties. Recently, Dey et al. (2019c) proposed AL Frechet distribution and studied its properties and real data application.

4.2 DUS Transformation

Kumar et al. (2015) proposed DUS transformation, which is defined as

$$ F(x)=\frac{e^{G(x) }-1}{e-1} \qquad x>0. $$

They also proposed DUS exponential distribution, by considering baseline CDF G(x) as exponential distribution which is parsimonious in parameter and having increasing hazard rate. Using the concept of DUS transformation, Maurya et al. (2017b) proposed DUS Lindley distribution by considering Lindley distribution as a baseline model. Based on this model, Tripathi et al. (2019) draw inferences on upper record values.

DUS exponential distribution can also be derived from the Poisson Family of distribution just by putting 𝜃 = 1 in Eq. 8. The DUS transformation can be derived by just putting 𝜃 = exp(1) = e in Eq. 3 in Poisson Family of distribution. Leahu et al. (2013) proposed the Poisson family of distribution the same as modified DUS transformation for the maximum case and Alkarni and Oraby (2012) for a minimum case, which is same as method given in equation Eq. 3 and Eq. 5 respectively. The DUS transformation and technique proposed by Alkarni and Oraby (2012) are the same, but only difference is that DUS uses the concept of Eq. 3 whereas other uses the concept of Eq. 5. Both techniques can also be interchangeable by just replacing 𝜃 = −𝜃.

4.3 Logarithmic Transformation

Motivated by Kumar et al. (2015), a new transformation was proposed by Maurya et al. (2016) and named it Logarithmic Transformation (LT) by taking logarithmic function instead of exponential function in DUS transformation. They also proposed LT exponential distribution by taking exponential baseline which has increasing hazard rate and parsimony in parameter. The LT method is given as;

$$ F(x)=1-\frac{\log(2-G(x))}{\log2} ;\qquad x>0. $$

LT method of proposing new distribution is independent to Pappas and Adamidis (2012) AL transformation, but can be obtained by just putting α = 2 in transformation. Latter, Goyal et al. (2020b) used this model to study Bayesian estimation under different loss functions.

4.4 GDUS Transformation

Maurya et al. (2017a) proposed a new transformation which is generalization of DUS transformation proposed by Kumar et al. (2015) and named it Generalized DUS (GDUS) transformation. They apply Lehmann type I on DUS transformation and defined as:

$$ F(x)=\frac{e^{G^{\alpha}(x) }-1}{e-1} \qquad x,\alpha>0. $$

The GDUS transformation can also be obtained by PF Family of distribution by just considering 𝜃 = 1 and generalized distribution function instead of only CDF. Latter, Goyal et al. (2020a) utilized this model to study Bayesian estimation under type-I progressive hybrid censoring. Based on the concept of GDUS transformation, Maurya et al. (2020) proposed GDUS Lindley distribution by considering Lindley distribution as baseline model.

4.5 Alpha Power Transformation

Mahdavi and Kundu (2017) proposed Alpha Power Transformation (APT). The CDF of the APT method is defined as

$$ F(x)=\frac{\alpha^{G(x) }-1}{\alpha-1} \qquad x, \alpha>0. $$

DUS transformation can also be derived by APT by considering α = exp(1) = e. Mahdavi and Kundu (2017) considered exponential baseline model and proposed alpha power exponential distribution. The APT can also be derived by just putting $\theta =\log (\alpha )$ in the Eq. 3. Based on APT, Nassar et al. (2017) proposed APT Weibull distribution and studied its statistical properties. Dey et al. (2017c) also proposed APT Weibull distribution and studied its statistical properties independently to Nassar et al. (2017), but both distributions are same in functional form. Dey et al. (2017a) studied properties of APT generalized exponential distribution. Latter, Nadarajah and Okorie (2018) provided closed form expressions for moment properties of this distribution and re-analyse with real data application. Dey et al. (2019a) proposed APT Lindley distribution, Elbatal et al. (2018) proposed a new transformation based on APT. Hassan et al. (2018) proposed APT extended exponential distribution, Hassan et al. (2019) proposed APT power Lindley distribution, Dey et al. (2019b) proposed APT inverse Lindley distribution. Jan et al. (2019) proposed alpha power generalized inverse Lindley distribution and studied its statistical properties and real life applications.

It is interesting that, Jones (2018) mentioned that this transformation can be obtained by transformation proposed by Barreto-Souza and Simas (2013) by just putting α = e^−λ. Also, the APT method can be derived by just putting $\theta =\log \alpha$ in Eq. 3 in PFD.

4.5.1 Extended Alpha Power Transformation

Ahmad et al. (2019) proposed extended APT by adding one more location parameter. The general form of the transformation is

$$ F(x)=\frac{\alpha^{G(x) }-e^{G(x) }}{\alpha-e} \qquad x, \alpha>e. $$

where e = exp(1). Actually, this transformation can be obtained by taking differences in numerator and denominator separately, between APT and DUS transformation. Based on this transformation, they proposed extended APT Weibull distribution and studies its properties and application.

4.6 Alpha New Logarithmic Transformation

Karakaya et al. (2017) proposed a new transformation named it Alpha New Logarithmic Transformation, which can be obtained by considering $\theta =\log \alpha$, in Poisson family of distribution and revert the transformation, (like wise Eq. 15). The transformation is

$$ F(x)=\frac{\log(1+\alpha G(x))}{\log(\alpha+1)} ;\qquad x, \alpha>0. $$

4.7 New Logarithmic Transformation

Maurya et al. (2018) proposed New Logarithmic Transformation (NLT). Actually, it was based on the LT method and Lehmann type II. The two transformations LT and NLT can be interchangeable by just considering Lehmann type I and II in one method. The NLT method is given as:

$$ F(x)=\frac{\log(1+G(x))}{\log2} ;\qquad x>0. $$

They considered exponential distribution as a baseline and proposed one parameter decreasing failure rate distribution and compare it with other famous decreasing failure rate models in different datasets of decreasing nature and found a better fit for others. If we consider $\theta =\log 2$, in Eq. 3 and revert the transformation, then the transformation can be obtained.

4.8 Generalized Logarithmic Transformation

Goyal et al. (2019) proposed a new transformation which is generalization of logarithmic transformation proposed by Maurya et al. (2016) and named it Generalized Logarithmic Transformation (GLT). They apply Lehmann type I on Logarithmic transformation and defined as:

$$ F(x)=1-\frac{\log[2- G^{\alpha}(x)] }{\log 2} \qquad x,\alpha>0. $$

The GLT can also be obtained by PF Family of distribution by reverting the Poisson Family of distribution in same fashion as in Logarithmic transformation method and applying the Lehmann type I.

4.9 Zubair Family of Distribution

Ahmad (2020) proposed Zubair family of distribution (ZFD) by his name and proposed Zubair Weibull distribution by just taking Weibull distribution as baseline model. The general form of new distribution is

$$ F(x)=\frac{e^{\theta G^{2}(x)}-1}{e^{\theta}-1} \qquad x, \theta>0. $$

Actually, this can be obtained by just putting α = 1 in PFD in Eq. 3. And for 𝜃 = 1, it become a particular case of Maurya et al. (2017a). Also, the proposed Zubair Weibull distribution is particular case of Mahmoudi and Jafari (2012), Mahmoudi and Shiran (2012), Leahu et al. (2013), and Ghorbani et al. (2014).

It is interesting that, Tahir and Cordeiro (2019) mentioned that this transformation can be obtained by transformation given in Eq. 9, proposed by Tahir and Cordeiro (2016) by just putting α = 2.

Table 5, gives a brief on transformation methods based on the Poisson family of distributions with the proposed year, title, and reference of the corresponding methods.

Table 5 Transformations based on Poisson family of distribution

5 Real data Application

In this section, we have considered three different real datasets of different nature for checking the suitability of the model in diverse fields. Further, to check the superiority of one model over other competitive models, we have considered eight different lifetime models which exhibit different type of hazard rates and are generated by Poisson family of distribution. All the considered models have either two or three parameters only. The first considered models are exponential Poisson (EP(max)) distribution (Cancho et al. 2011; Rezaei and Tahmasbi, 2012), generalized exponential Poisson (GEP(max)) distribution (Mahmoudi and Jafari, 2012), Weibull Poisson (WP(max)) distribution (Leahu et al. 2013), Lomax Poisson (LoP(max)) distribution (Al-Zahrani and Sagor, 2014), exponentiated exponential Poisson (EEP(min)) distribution (Barreto-Souza and Cribari-Neto, 2009), Weibull Poisson (WP(min)) distribution (Hemmati et al. 2011; Bereta et al. 2011; Lu and Shi, 2012), generalized exponential Poisson (GEP(min)) distribution (Ristić and Nadarajah, 2014) and Lindley Poisson (LP(min)) distribution (Gui et al. 2014). Here, the term used in bracket max or min used for notation, whether the concerned model comes from the concept of maximum or minimum, accordingly.

A brief description of the considered real datasets are given below:

Dataset 1: :

128 remission time of bladder cancer patient data. This dataset was reported by Lee and Wang (2003) and the most frequently used dataset. Five different authors used this dataset for seven models for real data illustration purposes.

Dataset 2: :

101 life of fatigue fracture of Kevlar 373/epoxy. This dataset was reported by Andrews and Herzberg (2012) and also most frequently used dataset. Six different authors used this dataset for five models for real data illustration purposes.

Dataset 3: :

30 successive values of March precipitation. This dataset was reported by Hinkley (1977) and also the most frequently used dataset. Two authors used this dataset for two models for real data illustration purposes.

To know the nature of the considered datasets, we have used the concept of scaled TTT plot, which shows that the Dataset 1 has UBT; Dataset 2 has BT; and Dataset 3 has IHR (see Fig. 1). And to check the model suitability criterion, we have considered four different criteria, namely AIC (Akaike Information Criterion), BIC (Bayesian information criterion), Kolmogorov—Smirnov (KS) test statistic and logarithmic of likelihood (Log L) value. The AIC, BIC and KS test statistic (D) are defined as;

$$ AIC=2*k-2*\log \hat{L}, \qquad BIC=k*\log(n)-2*\log \hat{L} , $$

$$ D=\underset{x}{Sup}~|F_{n}(x)-F(x)| , \qquad \text{where}\qquad F_{n}(x)=\frac{1}{n}\sum\limits_{i=1}^{n}I_{x_{i}\leq x} $$

is empirical distribution function, F(x) is the CDF, n is the size of the sample, k is a number of parameters involved in the model, and $\hat {L}$ is the maximum value of likelihood function for the taken model. In the KS test criterion, the value of test statistic and the corresponding p-value have been reported. The AIC and BIC are model selection criterion, but BIC gives penalties for involving more parameters in the model. It is obvious that the smaller the value of AIC, BIC, KS statistic, and the negative value of log-likelihood indicate the better fitting of the model.

Figure 1

Scaled TTT plots for considered datasets

The value of maximum likelihood estimates of the parameters, negative log-likelihood value (-Log L), KS distance (D) along with associated p-value (P), and model selection criterion i.e., AIC and BIC for considered models are reported in Table 6. All the values of the table are rounded up to four decimal places. From this table, one can conclude that;

Table 6 Table for ML estimates, KS statistics, AIC and BIC

Dataset 1: :

All the considered models fitted at 5% level of significance and KS distance is minimum for LoP(max) model with the minimum (-Log L) value and AIC value. But the BIC value is minim for LP(min) distribution because LP(min) is two-parameter while LoP(max) is a three-parameter model.

Dataset 2: :

Here, also, all the models fitted well to the dataset at the desired level of significance i.e., at 5%. Here, WP(max) model has a minimum KS distance with minimum negative log-likelihood value. But in the model selection criterion, AIC and BIC both are least for LP(min) model, as LP(min) has IHR and BT type hazard rates.

Dataset 3: :

Again, all the models fitted well at a 5% level of significance. Here, WP(min) model has a minimum KS distance with minimum negative log-likelihood value. But in the model selection criterion, AIC and BIC both are least for EP(max) model, as EP(max) is a two-parameter model having IHR.

From the above discussion, we can say that all the considered models are very flexible to fit a variety of datasets. However, according to model selection criterion i.e., AIC and BIC, LoP(max), LP(min), and EP(max) models are more flexible than others considered models for the selected datasets. We have also considered ECDF and fitted CDF plots for the considered datasets (see Fig. 2) and kernel density, fitted density and relative histogram plots for the considered datasets (see Fig. 3), to better visualization of the fitting of the models.

Figure 2

ECDF and fitted CDF plots for considered datasets

Figure 3

Kernel density, fitted density and relative histogram plots for considered datasets

6 Conclusion

In the present article, we have introduced a literature review of Poisson generated family of distribution and its associated distribution. There are several lifetime models, and transformation techniques are available in the literature that is related to this family, but most are not aware of it. This paper provided a complete list of the Poisson family of distribution year by year. Poisson family of distributions is more suitable in case of complementary and competitive risk and frequently occurs in reliability, data modeling, biological systems, health care field, and actuarial science. Also, the Poisson generated family of distributions provides a more flexible distribution than the existing baseline model and has the capability to fit a wide variety of skewed data. We have provided some basic distributional properties of proposed distributions as the nature of PDF and its limiting case, hazard rate, its sub–models, stochastic and likelihood order relationship, and detailed discussion about utilized real datasets; whatever the considered model discussed. Also, there are 40 datasets used by 54 authors in 62 models. We have also provided a section of real data analysis. In this section, we have considered three different real datasets and eight-lifetime models to check the applicability of these models. We have provided its parameter estimates, log-likelihood value, KS statistic along with p-value, and model section criterion as AIC and BIC. The fitting of these models over the datasets is also visualized by different plots as ECDF plots, kernel density plots along with fitted density and relative histogram plots.

In this context, we have listed 11 power series distributions by 12 different authors, 25 lifetime distributions by 19 authors based on the maximum concept in which two distributions (EP and EWP) have been proposed twice by different authors. Similarly, there are 52 distributions listed by 36 authors based on the minimum concept, in which WP distribution has been proposed five times by different authors. This means there are a total of 77 distributions proposed by 55 authors based on the Poisson family of distribution. In transformation methods, there are 23 distributions based on ten different transformation methods listed.

Now, if we count all the listed lifetime distributions based on the Poisson family of distribution, it reached nearly 122, which shows the role and importance of the Poisson family of distribution. In this way, we can say that this will be helpful to researchers in the field of lifetime distribution.