The Exponentiated Generalized-G Poisson Family of Distributions

1 Introduction

In the last decades, several generalized distributions have been proposed based on different modification methods. These modification methods require the addition of one or more parameters to base model which could provide better adaptability in the modeling of real lifetime data. Modern computing technology has made many of these techniques accessible even if analytical solutions are very complicated. Several continuous univariate-G families have recently appeared. Some notable family includes Marshall–Olkin-G family by Marshall and Olkin [22], exponentiated-G class by R. C. Gupta, P. L. Gupta and R. D. Gupta [17], transmuted exponentiated generalized-G family by Yousof, Afify, Alizadeh, Butt, Hamedani and Ali [36], transmuted geometric-G by Afify, Alizadeh, Yousof, Aryal and Ahmad [1], Kumaraswamy transmuted-G by Afify, Cordeiro, Yousof, Alzaatreh and Nofal [2], Burr X-G by Yousof, Afify, Hamedani and Aryal [37], the odd Lindley-G family of distributions by Silva, Percontini, de Brito, Ramos, Venancio and Cordeiro [33], exponentiated transmuted-G family by Merovci, Alizadeh, Yousof and Hamedani [23], the odd-Burr generalized family by Alizadeh, Cordeiro, Nascimento, Lima and Ortega [5], the transmuted Weibull-G family by Alizadeh, Rasekhi, Yousof and Hamedani [6], the type I half-logistic family by Cordeiro, Alizadeh and Diniz Marinho [11], the complementary generalized transmuted Poisson family by Alizadeh, Yousof, Afify, Cordeiro and Mansoor [7], the Zografos–Balakrishnan odd log-logistic family of distributions by Cordeiro, Alizadeh, Ortega and Serrano [12], logistic-X by Tahir, Cordeiro, Alzaatreh, Mansoor and Zubair [34], a new Weibull-G by Tahir, Zubair, Mansoor, Cordeiro, Alizadeh and Hamedani [35], the generalized odd log-logistic family by Cordeiro, Alizadeh, Ozel, Hosseini, Ortega and Altun [13], the beta odd log-logistic generalized family of distributions by Cordeiro, Alizadeh, Tahir, Mansoor, Bourguignon and Hamedani [14], beta transmuted-H by Afify, Yousof and Nadarajah [3], generalized transmuted-G by Nofal, Afify, Yousof and Cordeiro [30] and beta Weibull-G family by Yousof, Rasekhi, Afify, Ghosh, Alizadeh and Hamedani [38] among others.

In this paper we propose and study a generalized family of distribution using the genesis of Poisson distribution with the following motivation. Suppose that a system has N subsystems functioning independently at a given time where N has zero truncated Poisson (ZTP) distribution with parameter λ. It is the conditional probability distribution of a Poisson-distributed random variable, given that the value of the random variable is not zero. The probability mass function (pmf) of N is given by

Note that for ZTP variable the expected value and variance are respectively given by

and

Suppose that the failure time of each subsystem has the exponentiated Generalized-G (“EGG⁢(a,b)” for short) distribution defined by the cumulative distribution function (cdf) and probability density function(pdf) given by

and

respectively, where a>0 and b>0 are two additional shape parameters. Let Yi denote the failure time of the ith subsystem and let X=min⁡{Y1,Y2,…,YN}. Then the conditional cdf of X given N is

Therefore, the unconditional cdf of X, as described in [31], can be expressed as

The cdf in (1) is called the exponentiated generalized G Poisson (“EGGP”) family of distributions. The corresponding pdf is

For b=1 we have EGP class of distribution and for a=1 we have GGP class of distribution both of which are embedded in EGGP class.

Using the power series expansion of exp⁡(x), we express the pdf in (2) as

Using the series expansion (1-z)b-1=∑j=0∞(-1)j⁢Γ⁢(b)j!⁢Γ⁢(b-j)⁢zj the last equation can be expressed as

where

and

This is the Exp-G pdf with power parameter (k+1). By integrating (3), we obtain the mixture representation of F⁢(x) as

where Πk+1⁢(x) is the cdf of the Exp-G family with power parameter (k+1). Equation (4) reveals that the EGGP density function is a linear combination of Exp-G densities. Thus, some structural properties of the new family such as the ordinary and incomplete moments and the generating function can be immediately obtained from well-established properties of the Exp-G distributions.

The properties of Exp-G distributions have been studied by many authors in recent years, see Mudholkar and Srivastava [25] and Mudholkar, Srivastava and Freimer [26] for exponentiated Weibull (EW) distributions, R. C. Gupta, P. L. Gupta and R. D. Gupta [17] for exponentiated Pareto distributions, Gupta and Kundu [18] for exponentiated exponential distributions, Nadarajah and Kotz [29] for the exponentiated-type distributions, Nadarajah [27] for exponentiated Gumbel distributions, Shirke and Kakade [32] for exponentiated log-normal distributions and Nadarajah and Gupta [28] for exponentiated gamma distributions (EGa), among others.

The rest of the paper is outlined as follows. In Section 2 we provide the formulation of EGGP models for two special distributions. Mathematical properties of the EGGP model are discussed in Section 3. In Section 4 we discuss stress-strength models. The order statistics is discussed in Section 5. Parameter estimation procedures using method of maximum likelihood are presented in Section 6. Section 7 provides the application of the two generalized distributions to model real world data. Some concluding remarks are given in Section 8.

2 Special Models

The formulation provided in Section 1 can be used to generalize any classical probability distribution. For illustration purpose we will generalize the following two popular and versatile distributions, namely: the Weibull (W) distribution and the Pareto (Pa) distribution. The parameters of these models are positive real numbers. The pdf and cdf of these distributions are provided in Table 1.

2.1 The EGWP Distribution

The cdf and pdf of the EG-Weibull Poisson (EGWP) distribution are given, respectively, by

F⁢(x)=1-exp⁡(-λ⁢{1-exp⁡(-a⁢(α⁢x)β)}b)[1-exp⁡(-λ)]

and

f⁢(x)=a⁢b⁢λ⁢β⁢αβ⁢xβ-1[1-exp⁡(-λ)]⁢exp⁡(-a⁢(α⁢x)β)⁢{1-exp⁡(-a⁢(α⁢x)β)}b-1exp⁡(λ⁢{1-exp⁡(-a⁢(α⁢x)β)}b).

Plots of the pdf and cdf of the EGWP distribution are displayed in Figure 1 for some parameter values. As we shall see from the graphs EGWP distribution is more flexible compare to classical Weibull distribution.

Figure 1

Pdf (top) and cdf (bottom) of EGWP distribution.

2.2 The EGPaP Distribution

The cdf and pdf of the EG-Pareto Poisson(EGPaP) distribution (for x>θ) are, respectively, given by

F⁢(x)=1-exp⁡(-λ⁢{1-(θx)a⁢α}b)[1-exp⁡(-λ)]

and

f⁢(x)=a⁢b⁢λ⁢α⁢θα⁢(θx)(a-1)⁢α⁢[1-(θx)a⁢α]b-1xα+1⁢[1-exp⁡(-λ)]⁢exp⁡(λ⁢[1-(θx)a⁢α]b).

Plots of the pdf and cdf of the EGPaP distribution are displayed in Figure 2 for some parameter values. As we shall see from the graphs EGPaP distribution is more flexible compare to the Pareto distribution.

Figure 2

Pdf (top) and cdf (bottom) of EGPaP distribution

3 Mathematical Properties

In this section we provide some structural and mathematical properties of the EGGP distribution including the quantile function, moments, entropy measure, residual and revered residual life.

4 Stress-Strength Models

The stress-strength model is the most widely used approach for reliability estimation. This model is used in many applications of physics and engineering, such as strength failure and system collapse. In stress-strength modeling,

is a measure of reliability of the system when it is subjected to random stress X2 and has strength X1 (see [20]). The system fails if and only if the applied stress is greater than its strength and the component will function satisfactorily whenever X1>X2. Moreover, R can be considered as a measure of system performance and naturally arises in electrical and electronic systems. Another interpretation can be that the reliability, say R, of the system is the probability that the system is strong enough to overcome the stress imposed on it. Let X1 and X2 be two independent random variables have EGGP⁢(x;a1,b1,λ1,𝝍) and EGGP⁢(x;a2,b2,λ2,𝝍) distributions. The reliability R is given by

Then

where

5 Order Statistics

Let X1,…,Xn be a random sample from the EGGP family of distributions and let X(1),…,X(n) be the corresponding order statistics. The pdf of the ith order statistic, say Xi:n, can be written as

where B⁢(⋅,⋅) is the beta function. Substituting (1) and (2) into equation (9), we get

where

Moreover, the pdf of Xi:n can be expressed as

therefore, the density function of the EGGP order statistics is a mixture of EG densities. Based on the last equation, we note that the properties of Xi:n follow from those properties of Yk+1. For example, the moments of Xi:n can be expressed as

For the EGWP model we have

The L-moments are analogous to the ordinary moments but can be estimated by linear combinations of order statistics. They exist whenever the mean of the distribution exists, even though some higher moments may not exist, and are relatively robust to the effects of outliers. Based upon the moments in equation (10), we can derive explicit expressions for the L-moments of X. They are linear functions of expected order statistics defined by

The first four L-moments are given by

One can simply obtain the L-moments λr for X from (10) with q=1.

6 Estimation

Let X1,…,Xn be a random sample from the EGGP distribution with parameters λ,a,b and 𝝍. Further, let 𝚿=(a,b,λ,𝝍⊺)⊺ be a ((p+3)×1) parameter vector, where 𝝍 is a (p×1) baseline parameter vector. For determining the MLE of 𝚿, we have the log-likelihood function

where G¯⁢(xi;𝝍)=1-G⁢(xi;𝝍),si=1-G¯⁢(xi;𝝍)a. The components of the score vector

are given by