A New Family of Continuous Probability Distributions

In this paper, a new parametric compound G family of continuous probability distributions called the Poisson generalized exponential G (PGEG) family is derived and studied. Relevant mathematical properties are derived. Some new bivariate G families using the theorems of “Farlie-Gumbel-Morgenstern copula”, “the modified Farlie-Gumbel-Morgenstern copula”, “the Clayton copula”, and “the Renyi’s entropy copula” are presented. Many special members are derived, and a special attention is devoted to the exponential and the one parameter Pareto type II model. The maximum likelihood method is used to estimate the model parameters. A graphical simulation is performed to assess the finite sample behavior of the estimators of the maximum likelihood method. Two real-life data applications are proposed to illustrate the importance of the new family.


Introduction and Genesis
In statistical literature, we always assume that every real phenomenon can be modeled by some lifetime distributions. If we know this distribution(s), we can then analyze our phenomenon, as many lifetime distributions have been developed in this regard. The well-known Poisson distribution is one of the famous distributions that was also defined and studied as a new family of continuous distribution in the concept of compounding. Using the Poisson G family, several compound lifetime G families have been proposed and studied. In the compounding method, there are two different approaches available; one is by using zero truncated power series (ZTPS) distribution and the other by using zero truncated Poisson (ZTP) distribution directly with other continuous distributions. A comprehensive survey regarding the Poisson G models was recently proposed by [1].
In this paper, we propose and study a new family of distributions using ZTP distribution with a strong physical motivation. Suppose that a system has N (a discrete random variable) subsystems functioning independently at a given time, where N has a ZTP distribution with parameter λ and the failure time of ith component Y i |i = 1, 2, . . . (say) is independent of N. It is the conditional probability distribution of a Poisson-distributed random variable (RV), given that the value of the RV is not zero. The probability mass function (PMF) of N is given by Then, the conditional CDF of X given N is Therefore, the unconditional CDF of X, as described in [3][4][5][6][7][8][9], can be expressed as The CDF in (2) is called the Poisson generalized exponential G (PGEG) family, V = (λ, θ, β, ξ) is the parameter vector of the PGE-G family. The corresponding PDF can be derived as where the function h θ,β,ξ (x) = dH θ,β,ξ (x)/dx. Or, the PDF due to (3) can be re-expressed as A RV X having PDF (4) is denoted by X ∼ PGE-G (V). Some special cases of the PGE-G family are listed in Table 1. Table 1. Some new members derived based on the Poisson generalized exponential G (PGEG) family.
2-The real datasets that have no extreme values, as shown in Section 6. 3-The real datasets whose nonparametric Kernel density is symmetric, as given in Section 6 ( Figure 11).
The PGE-G family proved its superiority against many well-known families as shown below: 1-In modeling the failure times of the aircraft windshield, the PGE-G family is better than the special generalized mixture G family, the odd log-logistic G family, the Burr-Hatke G family, the transmuted Topp-Leone G family, the Gamma G family, the Kumaraswamy G family, the McDonald G family, the exponentiated G family, and the proportional reversed hazard rate G family under the Akaike information criteria, consistent information criteria, Bayesian information criteria, and Hannan-Quinn information criteria.
2-In modeling the service times of the aircraft windshield, the PGE-G family is better than the special generalized mixture G family, the odd log-logistic G family, the Burr-Hatke G family, the transmuted Topp-Leone G family, the Gamma G family, the Kumaraswamy G family, the McDonald G family, the exponentiated G family, and the proportional reversed hazard rate G family under the Akaike information criteria, consistent information criteria, Bayesian information criteria, and Hannan-Quinn information criteria.

BvPGE-G Type via CCp
Let us assume that X 1 ∼ PGE-G(V 1 ) and X 2 ∼ PGE-G(V 2 ). The CCp depending on the continuous marginal functions w = 1 − w and = 1 − can be considered as Then, the BvPGE-G-type distribution can be obtained from (5). A straightforward multivariate PGE-G (m-dimensional extension) via CCp can be easily derived analogously. The m-dimensional extension via CCp is a function operating in [0, 1] m , and in that case, x i is not a value in [0, 1] necessarily.

BvPGE-G Type via RECp
Following [18], the RECp can be derived as C(w, ) = x 2 w + x 1 − x 1 x 2 , with the continuous marginal functions w = 1 − w = F V 1 (x 1 ) ∈ (0, 1) and = 1 − = F V 1 (x 2 ) ∈ (0, 1), where the values x 1 and x 2 are in order to guarantee that C(w, ) is of a copula. Then, the associated CDF of the BvPGE-G will be where F V i (x i ) is defined above. It is worth mentioning that in [18], the authors emphasize that this copula does not show a closed shape and numerical approaches become necessary.

BvPGE-G Type via Modified FGMCp
The modified formula of the modified FGMCp due to [17] can written as Then, for 1 ≤ min(βα, ηξ), we have The following four types can be derived and considered: • Type I Let H 1 (w) = λ 1 H θ 1 ,β 1 ,ξ (w) and H 2 ( ) = λ 2 H θ 2 ,β 2 ,ξ ( ). Then, the new bivariate version via modified FGMCp type I can be written as Then, the corresponding bivariate version (modified FGMCp Type II) can be derived from Then, the associated CDF of the BvPGE-G-FGM (modified FGMCp Type III) can be written as

• Type IV
Using the quantile concept, the CDF of the BvPGE-G-FGM (modified FGMCp Type IV) model can be obtained using

BvPGE-G Type via Ali-Mikhail-Haq Copula
Under the "stronger Lipschitz condition" and following [19], the joint CDF of the Archimedean Ali-Mikhail-Haq copula can written as and the corresponding joint PDF of the Archimedean Ali-Mikhail-Haq copula can be expressed as

Expanding the Univariate PDF
In this subsection, we present a useful representation for the new PDF in (4). Based on the new representation, we can easily and directly derive the main statistical properties of the new family due to the exponentiated G (exp-G) family. Using the power series, we expand the quantity A λ,θ,β (x). Then, the PDF in (4) can be expressed as Considering the power series and applying (7) to the quantity B θ( +1),β (x) in (6), we get Inserting the above expression of C β( +1) (x) in (8), the PGE-G density reduces to via generalized binomial expansion, we get Inserting (10) in (9), the PGE-G density can be expressed as which is an infinite linear combination of exp-G PDFs where g κ * (x) = dG κ * (x)/dx = κ * π(x)G ξ (x) κ+j is the PDF of the exp-G family with power k * and υ κ,j is a constant where Similarly, the CDF of the PGE-G family can also be expressed as where G k * (x) is the CDF of the exp-G family with power k * .

Convex-Concave Analysis
Convex PDFs play a very important role in many areas of mathematics. They are important especially in study of the "optimization problems" where they are distinguished by several convenient properties. In mathematical analysis, a certain PDF defined on a certain n-dimensional interval is called "convex PDF" if the line between any two points on the graph of the PDF lies above the graph between the two points.
The PDF in (4) and based on any base-line model (see Table 1) is said to be "concave PDF" if for any The PDF in (4) is said to be "convex PDF" if for any , then c f ∆x 1 + ∆x 2 is convex for every c > 0. If f ∆x 1 + ∆x 2 and g ∆x 1 + ∆x 2 are "convex PDF", then f ∆x 1 + ∆x 2 + g ∆x 1 + ∆x 2 is also "convex PDF".

Moments
Let Y κ * be an RV having the exp-G family power with k * and X be an RV having the PGE-G family. Then, the rth moment of the RV X is Analogously, the nth moment around the arithmetic mean (µ 1 ) of X is

Moment-Generating Function (MGF)
We present two formulas for the obtaining the MGF. Clearly, the first formula can be derived from Equation (11) as where M κ * (t) is the MGF of the RV Y κ * . However, the second formula is based on the concept of the quantile function (QF) as can be numerically evaluated using the baseline QF, i.e., Q G (u) = G −1 (u).

Incomplete Moments (IM)
The sth IM, say φ s,X (t), of the RV X can be derived from (11) as φ s, One of the main mathematical end economical applications of the first IM concerns "mean deviations (MD)" and "Bonferroni and Lorenz curves", which are very useful in economics, insurance, demography, reliability, and medicine. The MD about the µ 1 of E X − µ 1 = a 1 , and the MD about the median (M) of E(|X − M|) = a 2 of the RV X are given by a 1 = 2µ 1 F µ 1 − 2φ 1,X µ 1 and Now, we provide two ways to determine a 1 and a 2 . First, φ 1, These results for φ 1,X (t) can be directly applied for calculating the Bonferroni and Lorenz curves defined, for a certain given probability P, by B(P ) = φ 1,X (Q(P ))/ P µ 1 and L(P ) = φ 1,X (Q(P ))/µ 1 , respectively.

Residual Life (RL) and Reversed Residual Life (RRL)
The q th moment of the RL of the RV X can be obtained from m q,X (t) = E[(X − t) q ] X>t and q∈N or from which can also be written as For q = 1, we obtain the mean of the residual life (MRL) also called the life expectation (LE), which can be drived from m 1,X (t) = E[(X − t)]| X>t and q∈N and represents the additional expected life for a certin system or component that is already alive at the age t. On the other hand, the q th moment of the RRL is M q,X (t) = E (t − X) q X≤t, t>0 and q∈N or which can also be expressed as For q = 1, we obtain the mean waiting time (MWT), which is also called the mean inactivity time (MIT), which can be derived from M 1,X (t) = E[(t − X)]| X≤t, t>0 and q=1 .

Mathematical Results and Numerical Analysis for Two Special Models
We present some mathematical results for two special models chosen from Table 1. All results listed in Table 2 were derived based on the mathematical results previously obtained in Sections 1-6.
the lower incomplete gamma function , and the upper incomplete gamma function

Property
Result Support  Table 3 below gives numerical analysis for the mean (E(X)), variance (V(X)), skewness (S(X)), and kurtosis (K(X)) for PGEPII model based on special case number 7 of Table 1 with a = 1. Based on results listed in Table 3, it is noted that E(X) decreases as λ increases, S(X) ∈ (0.647392, ∞) and K(X) ranging from 5.07 to ∞.

The Maximum Likelihood Estimation (MLE) Method
Let x 1 , . . . , x n be an observed random sample (RS) from the PGE-G family with V = λ, θ, β, ξ T T . The function of the log-likelihood can be obtained and maximized directly using the R software (the "optim function") or the program of Ox (sub-routine of MaxBFGS) or MATH-CAD software or by solving the nonlinear equations of the likelihood derived from differentiating V . The score vector components V can be easily derived from obtaining the nonlinear system U λ = U θ = U β = U ξ k = 0 and then simultaneously solving them for getting the MLE of V. This system could be solved numerically for the complicated models using iterative algorithms such as the "Newton-Raphson" algorithms. We can compute the maximum values of the unrestricted and restricted log-likelihoods to obtain likelihood ratio (LR) statistics for testing some sub models. Hypothesis tests of the type H 0 : Ω = Ω 0 versus H 1 : Ω = Ω 0 , where Ω is a vector formed with some components of V and Ω 0 is a specified vector, can be performed using LR statistics. For example, the test of H 0 : λ = θ = β = 1 versus H 1 : H 0 is not true and is equivalent to comparing the PGE-G and G distributions, and the LR statistic is given by W LR = 2{ V ( λ, θ, β, ξ T ) − (1, 1, 1, ξ T )}, where λ, θ, β and ξ T are the MLEs under H and ξ T is the estimate under H 0 .

Graphical Assessment
We present a graphical simulation for assessing the behavior of the finite sample of the MLEs for the PGEPII distribution. We maximized the log-likelihood function using a wide range of starting initial values. The starting initial values were taken in a fine scale. For the PGEPII model, for example, they were taken corresponding to all possible combinations of λ = 1, 2, . . . , 100, θ = 1, 2, . . . , 100, β = 1, 2, . . . , 100, and c = 1, 2, . . . , 100. The proposed assessment is performed depending on the following assessing algorithm: Using the QF of the PGEPII distribution, we generate 1000 samples of size n from the PGEPII and PGEE models where Computing the standard errors (SEs) of the MLEs for the N = 1000 samples, SEs are obtained via inverting the "observed information matrix".
For PGEPII model,

versus
: ≠ , where is a vector formed with some components of and is a specified vector, can be performed using LR statistics. For example, the test of : = = = 1 versus : is not true and is equivalent to comparing the PGE-G and G distributions, and the LR statistic is given by = 2{ℓ ( , , , ) − ℓ(1,1,1, )}, where , , and are the MLEs under and is the estimate under .

Graphical Assessment
We present a graphical simulation for assessing the behavior of the finite sample of the MLEs for the PGEPII distribution. We maximized the log-likelihood function using a wide range of starting initial values. The starting initial values were taken in a fine scale. For the PGEPII model, for example, they were taken corresponding to all possible combinations of = 1,2, … ,100, = 1,2, … ,100, = 1,2, … ,100, and = 1,2, … ,100 . The proposed assessment is performed depending on the following assessing algorithm: Using the QF of the PGEPII distribution, we generate 1000 samples of size from the PGEPII and PGEE models where Computing the standard errors (SEs) of the MLEs for the N = 1000 samples, SEs are obtained via inverting the "observed information matrix".

Modeling Failure and Service Times
Two real-life data applications to illustrate the importance and flexibility of the family are presented under the PII case. The fits of the PGEPII are compared with other PII models shown in Table 4.

Modeling Failure and Service Times
Two real-life data applications to illustrate the importance and flexibility of the family are presented under the PII case. The fits of the PGEPII are compared with other PII models shown in Table 4.
The first dataset (aircraft windshield, n = 84): The first real-life dataset represents the data on failure times of 84 aircraft windshield. The second dataset (aircraft windshield, n = 63): The second real-life dataset represents the data on service times of 63 aircraft windshield. The two real-life datasets were chosen based on matching their characteristics and the plots of the PDF in Figure 1 (the right panel). By examining Figure 1 (the right panel), it is noted that the new PDF can be asymmetric right-skewed function" and "symmetric" with different shapes. On the other hand, by exploring the two real datasets, it is noted that densities are nearly symmetric functions. Additionally, the HRF of the new family includes the asymmetric monotonically increasing shape, and the HRF of the two real datasets are asymmetric monotonically increasing (see Figure 1(left panel)). The two real datasets were reported by [20]. Many other symmetric and asymmetric useful real-life datasets can be found in [21][22][23][24][25][26][27][28]. Initial density shape is explored using the nonparametric "Kernel density estimation (KDE)" approach in Figure 7. The "normality" condition is checked via the "quantile-quantile (Q-Q) plot" in Figure 8. The initial shape of the empirical HRFs is discovered from the "total time in test (TTT)" plot in Figure 9. The extremes are spotted from the "box plot" in Figure 10. Based on Figure 7, it is noted that the densities are nearly symmetric functions. Based on Figure 8, we see that the "normality" nearly exists. Based on Figure 9, it is noted that the HRF is "asymmetric monotonically increasing shaped" for the two datasets. Based on Figure 10, it is showed that no extreme observations were founded. The goodness-of-fit (GOF) statistic called "Akaike information" (AICr), consistent-AIC (CAICr), Bayesian-IC (BICr), and Hannan-Quinn-IC (HQICr) were analyzed by comparing the competitive PII models. However, many other PII extensions could be considered in comparisons [37][38][39][40][41][42][43][44][45]. For failure times real-life data, relevant numerical results are shown in Tables 5 and 6. Precisely, Table 5 gives the MLEs and SEs. Table 6 gives the four GOF test statistics. For service times real-life data, the results are presented in Tables 7 and 8. Precisely, Table 5 gives the MLEs and SEs, whereas Table 8 gives the four GOFs test statistics. Figures 11 and 12 give the probability-probability (P-P) plot, estimated PDF (EPDF), Kaplan-Meier survival (KMS) plot and estimated HRF (EHRF) plot for the two datasets, respectively. Based on Tables 6  and 8 vice times real-life data, the results are presented in Tables 7 and 8. Precisely, Table 5 gives the MLEs and SEs, whereas Table 8 gives the four GOFs test statistics. Figures 11 and 12 give the probability-probability (P-P) plot, estimated PDF (EPDF), Kaplan-Meier survival (KMS) plot and estimated HRF (EHRF) plot for the two datasets, respectively. Based on Table 6 and   the MLEs and SEs, whereas Table 8 gives the four GOFs test statistics. Figures 11 and 12 give the probability-probability (P-P) plot, estimated PDF (EPDF), Kaplan-Meier survival (KMS) plot and estimated HRF (EHRF) plot for the two datasets, respectively. Based on Table 6 and                The results of the LR statistics of the PGEPII model against the QPGEPII, PEPII, and QPPII models under the second dataset are in Table 10. Based on the results of this table, I-We reject the null hypotheses of the LR tests in favor of the PGEPII model. II-We can confirm the significance of the parameters λ and θ with W LR = 33.01982, W LR = 4.710811, and W LR = 3.476109, respectively.   Figure 12. EPDF, EHRF, P-P, and KMS plots for the 2nd dataset.

Conclusions
In this article, a new parametric lifetime compound G family of continuous probability distributions called the Poisson generalized exponential G (PGEG) family is derived and studied. The PGEG family is defined based on the Poisson and the generalized exponential G families' concept of compounding. The new density can be "asymmetric rightskewed function", "asymmetric left-skewed", "bimodal", and "symmetric" with different shapes. The new HRF can be "upside down bathtub", "bathtub", "decreasing-constant", "increasing-constant", "increasing", "constant", and "increasing". Relevant mathematical properties including moments, incomplete moments, and mean deviation are derived. Some new bivariate-type PGEG families using the "copula of Farlie-Gumbel-Morgenstern", "copula of the modified Farlie-Gumbel-Morgenstern", "the Clayton copula", and "copula Renyi's entropy" are presented. Many special members are derived, and special

Conclusions
In this article, a new parametric lifetime compound G family of continuous probability distributions called the Poisson generalized exponential G (PGEG) family is derived and studied. The PGEG family is defined based on the Poisson and the generalized exponential G families' concept of compounding. The new density can be "asymmetric rightskewed function", "asymmetric left-skewed", "bimodal", and "symmetric" with different shapes. The new HRF can be "upside down bathtub", "bathtub", "decreasing-constant", "increasing-constant", "increasing", "constant", and "increasing". Relevant mathematical properties including moments, incomplete moments, and mean deviation are derived. Some new bivariate-type PGEG families using the "copula of Farlie-Gumbel-Morgenstern", "copula of the modified Farlie-Gumbel-Morgenstern", "the Clayton copula", and "copula Renyi's entropy" are presented. Many special members are derived, and special attention is devoted to the exponential (E) and the one parameter Pareto type II (PII) model. A simulation study is presented to assess the finite sample behavior of the estimators. The simulations are based on a certain given algorithm under the baseline PII model. Finally, two different real-life applications are proposed to illustrate the importance of the PGEG family. For all real data, for exploring the "initial shape", the nonparametric Kernel density estimation is presented. For checking the "normality" condition, the "Quantile-Quantile plot" is presented. For discovering the shape of the HRFs, the "total time in test" plot is provided. To explore the extremes, the "box plot" is sketched. Data Availability Statement: The two real datasets were reported by [20].