A Generalization of Binomial Exponential-2 Distribution: Copula, Properties and Applications

In this paper, we propose a new three-parameter lifetime distribution for modeling symmetric real-life data sets. A simple-type Copula-based construction is presented to derive many bivariate- and multivariate-type distributions. The failure rate function of the new model can be “monotonically asymmetric increasing”, “increasing-constant”, “monotonically asymmetric decreasing” and “upside-down-constant” shaped. We investigate some of mathematical symmetric/asymmetric properties such as the ordinary moments, moment generating function, conditional moment, residual life and reversed residual functions. Bonferroni and Lorenz curves and mean deviations are discussed. The maximum likelihood method is used to estimate the model parameters. Finally, we illustrate the importance of the new model by the study of real data applications to show the flexibility and potentiality of the new model. The kernel density estimation and box plots are used for exploring the symmetry of the used data.


Introduction
The monotonicity asymmetric failure (hazard) rate function (HRF) of a certain lifetime probabilistic distribution has an important role in modeling real lifetime data. Distributions with the "monotonicity increasing" failure rate (MIFR) function have useful real applications in "pricing" and "supply" chain contracting problems. The MIFR property is a well-known and useful concept in "dynamic programming", "reliability theory" and other areas of applied probability and statistics (see [1,2]). The paper [3] introduced a new two-parameter lifetime model with MIFR named the binomial-exponential-2 (BE2) model, which is constructed as a model of a random sum (RSm) of independent exponential random variables (RVs) when the sample size has a "zero truncated binomial" distribution. The BE2 distribution can be used as an alternative to the Weibull (W), gamma (Gam), exponentiated exponential (EE), and weighted exponential (WhE) distributions in real life applications.
The BE2 model is a mixture of the standard exponential (with parameter α) model and standard gamma model (with shape parameter 2 and scale parameter α); when θ = 0, we get the standard exponential model, and when θ = 1, the BE2 model reduces to the Gam model. In the last few decades, many new G families of continuous distributions have been developed. One of the most famous ones is called the new type II half-logistic (TIIHL-G) family (see [4]). According to [4], the CDF of the TIIHL-G family of distributions is given by where G Ψ (y) is the baseline CDF depending on a parameter vector Ψ and λ > 0 is an additional shape parameter. For each baseline G Ψ (y), we can generate a new TIIHL model using (4). The corresponding PDF to (4) is given by f λ,Ψ (y) = 2λ g Ψ (y)G Ψ (y) λ−1 where g Ψ (y) = dG Ψ (y)/dx is the baseline PDF. Equation (5) will be most tractable when G Ψ (y) and g Ψ (y) have simple expressions. The survival function, the failure (hazard) rate function and the quantile function are F λ,Ψ (y) = 1−G Ψ (y) λ 1+G Ψ (y) λ , h λ,Ψ (y) = 2λg Ψ (y)G Ψ (y) λ−1 1−G Ψ (y) 2λ , and Q(u) = G −1 λ u 2−u . Equations (4) and (5) are used for generating the new model.

The New Model and Its Motivation
In this section, we introduce the three-parameter type II half-logistic binomial exponential 2 (TIIHLBE2) distribution. Substituting from (1) into (4), the CDF of the TIIHLBE2 (or expanded BE2 "EBE" for short) model can be expressed as The corresponding PDF is given by Here and henceforth, an RV Y having PDF (7) is denoted by Y ∼ EBE (λ, α, θ). For the EBE distribution, the HRF can be derived as h λ,α,θ (y) = 2λαe −αy 1 + (αy−1)θ 2−θ 1 − 1 + θαy 2−θ e −αy 1 + 1 − 1 + θαy 2−θ e −αy λ . Figure 1 presents some plots of the PDF of the EBE model for some different values of the parameters λ, α and θ. We note that the new PDF can be "right skewed" with different shapes of "skewness" and "kurtosis". Here and henceforth, an RV having PDF (7) is denoted by ∼EBE ( , , ). For the EBE distribution, the HRF can be derived as ℎ , , ( ) = 2 1 + ( − 1) 2 − 1 − 1 + 2 − 1 + 1 − 1 + 2 − . (8) Figure 1 presents some plots of the PDF of the EBE model for some different values of the parameters , and . We note that the new PDF can be "right skewed" with different shapes of "skewness" and "kurtosis".  Figure 2 gives the plots of the HRF of the EBE distribution. We note that the new HRF can be "increasing", "increasing-constant", "decreasing" and "upside-down-constant" shaped. Thus, the new model may be useful in modeling different shapes of real data.  Figure 2 gives the plots of the HRF of the EBE distribution. We note that the new HRF can be "increasing", "increasing-constant", "decreasing" and "upside-down-constant" shaped. Thus, the new model may be useful in modeling different shapes of real data. Here and henceforth, an RV having PDF (7) is denoted by ∼EBE ( , , ). For the EBE distribution, the HRF can be derived as ℎ , , ( ) = 2 1 + ( − 1) 2 − 1 − 1 + 2 − 1 + 1 − 1 + 2 − . (8) Figure 1 presents some plots of the PDF of the EBE model for some different values of the parameters , and . We note that the new PDF can be "right skewed" with different shapes of "skewness" and "kurtosis".  Figure 2 gives the plots of the HRF of the EBE distribution. We note that the new HRF can be "increasing", "increasing-constant", "decreasing" and "upside-down-constant" shaped. Thus, the new model may be useful in modeling different shapes of real data. ee [6][7][8][9][10][11][12].

BivEBE Type via "Modified FGM" (MFGM) Copula
The modified JCDF of the bivariate FGM copula can be expressed as

BivEBE Type via "Modified FGM" (MFGM) Copula
The modified JCDF of the bivariate FGM copula can be expressed as

6-12].
ied FGM" (MFGM) Copula the bivariate FGM copula can be expressed as where ( ) and ( ) are two absolutely continuous ) Model e-I) model can be derived directly using The unknown parameter ∆ is a dependence parameter, and for every er the EBE Model E (BivEBE) Type via Renyi's Entropy [5], the joint CDF (JCDF) of the "Renyi's entropy Copula" can be expressed as + − ; then, the associated BivEBE will be ( , ) = ( ( ), ( )) are the parameter vectors for ( ) and ( ), respectively.

GM (Type-I) Model
E-FGM (Type-I) model can be derived directly using

BivEBE Type via "Modified FGM" (MFGM) Copula
The modified JCDF of the bivariate FGM copula can be expressed as

BivEBE-FGM (Type-I) Model
The BivEBE-FGM (Type-I) model can be derived directly using

odified FGM" (MFGM) Copula
F of the bivariate FGM copula can be expressed as

BivEBE Type via "Modified FGM" (MFGM) Copula
The modified JCDF of the bivariate FGM copula can be expressed as

BivEBE-FGM (Type-I) Model
The BivEBE-FGM (Type-I) model can be derived directly using

BivEBE Type via "Modified FGM" (MFGM) Copula
The modified JCDF of the bivariate FGM copula can be expressed as

BivEBE Type via "Modified FGM" (MFGM) Copula
The modified JCDF of the bivariate FGM copula can be expressed as

BivEBE-FGM (Type-I) Model
The BivEBE-FGM (Type-I) model can be derived directly using

BivEBE Type via "Modified FGM" (MFGM) Copula
The modified JCDF of the bivariate FGM copula can be expressed as

BivEBE-FGM (Type-I) Model
The BivEBE-FGM (Type-I) model can be derived directly using

BivEBE Type via "Modified FGM" (MFGM) Copula
The modified JCDF of the bivariate FGM copula can be expressed as

BivEBE Type via "Modified FGM" (MFGM) Copula
The modified JCDF of the bivariate FGM copula can be expressed as Copula under the EBE Model

BivEBE Type via "Modified FGM" (MFGM) Copula
The modified JCDF of the bivariate FGM copula can be expressed as

BivEBE Type via "Modified FGM" (MFGM) Copula
The modified JCDF of the bivariate FGM copula can be expressed as Then,

BivEBE Type via "Modified FGM" (MFGM) Copula
The modified JCDF of the bivariate FGM copula can be expressed as Then, where O( ].

BE Type via "Modified FGM" (MFGM) Copula
e modified JCDF of the bivariate FGM copula can be expressed as

. BivEBE Type via "Modified FGM" (MFGM) Copula
The modified JCDF of the bivariate FGM copula can be expressed as Then,

BivEBE Type via "Modified FGM" (MFGM) Copula
The modified JCDF of the bivariate FGM copula can be expressed as Then,

. BivEBE Type via "Modified FGM" (MFGM) Copula
The modified JCDF of the bivariate FGM copula can be expressed as

BivEBE Type via "Modified FGM" (MFGM) Copula
The modified JCDF of the bivariate FGM copula can be expressed as

BivEBE Type via "Modified FGM" (MFGM) Copula
The modified JCDF of the bivariate FGM copula can be expressed as

BivEBE Type via "Modified FGM" (MFGM) Copula
The modified JCDF of the bivariate FGM copula can be expressed as

BivEBE Type via "Modified FGM" (MFGM) Copula
The modified JCDF of the bivariate FGM copula can be expressed as

BivEBE Type via "Modified FGM" (MFGM) Copula
The modified JCDF of the bivariate FGM copula can be expressed as

BivEBE Type via "Modified FGM" (MFGM) Copula
The modified JCDF of the bivariate FGM copula can be expressed as

BivEBE Type via "Modified FGM" (MFGM) Copula
The modified JCDF of the bivariate FGM copula can be expressed as

BivEBE Type via "Modified FGM" (MFGM) Copula
The modified JCDF of the bivariate FGM copula can be expressed as

BivEBE Type via "Modified FGM" (MFGM) Copula
The modified JCDF of the bivariate FGM copula can be expressed as

BivEBE Type via "Modified FGM" (MFGM) Copula
The modified JCDF of the bivariate FGM copula can be expressed as

BivEBE Type via "Modified FGM" (MFGM) Copula
The modified JCDF of the bivariate FGM copula can be expressed as

EBE Type via "Modified FGM" (MFGM) Copula
e modified JCDF of the bivariate FGM copula can be expressed as

BivEBE Type via "Modified FGM" (MFGM) Copula
The modified JCDF of the bivariate FGM copula can be expressed as

BivEBE-FGM (Type-I) Model
The BivEBE-FGM (Type-I) model can be derived directly using In this case, one can also derive a closed form expression for the associated CDF of the BivEBE-FGM (Type-III).

. BivEBE Type via "Modified FGM" (MFGM) Copula
The modified JCDF of the bivariate FGM copula can be expressed as The BivEBE-FGM (Type-I) model can be derived directly using

BivEBE Type via "Modified FGM" (MFGM) Copula
The modified JCDF of the bivariate FGM copula can be expressed as The BivEBE-FGM (Type-I) model can be derived directly using

BivEBE Type via "Modified FGM" (MFGM) Copula
The modified JCDF of the bivariate FGM copula can be expressed as The BivEBE-FGM (Type-I) model can be derived directly using

BivEBE Type via "Modified FGM" (MFGM) Copula
The modified JCDF of the bivariate FGM copula can be expressed as

BivEBE Type via "Modified FGM" (MFGM) Copula
The modified JCDF of the bivariate FGM copula can be expressed as

BivEBE Type via "Modified FGM" (MFGM) Copula
The modified JCDF of the bivariate FGM copula can be expressed as

BivEBE Type via "Modified FGM" (MFGM) Copula
The modified JCDF of the bivariate FGM copula can be expressed as

BivEBE Type via "Modified FGM" (MFGM) Copula
The modified JCDF of the bivariate FGM copula can be expressed as

BivEBE Type via "Modified FGM" (MFGM) Copula
The modified JCDF of the bivariate FGM copula can be expressed as

BivEBE Type via "Modified FGM" (MFGM) Copula
The modified JCDF of the bivariate FGM copula can be expressed as Then, Then, the BivEBE-type distribution can be derived from F(t, w) = H F V 1 (t), F V 2 (w) . A straightforward n-dimensional extension from the above will be H (

BivEBE Type via "Modified FGM" (MFGM) Copula
The modified JCDF of the bivariate FGM copula can be expressed as Then, Symmetry 2020, 8, x FOR PEER REVIEW 4 of 17

Moments
Theorem 1. If Y ∼EBE (λ, α, θ), then the r th moment of Y is given by where Proof. Let Y be an RV following the EBE distribution. The r th ordinary moment can be obtained using the well-known formula µ r (y) = (1 − θ)y r+κ + θαy r+κ+1 e −α(1+ j)y dy.
If we set r = 1, we obtain the mean of the EBE distribution. Variance, skewness and kurtosis measures can be easily derived from the well-known relationships. Three-dimensional plots of the skewness and kurtosis of the EBE model are presented in Figures 3 and 4. If we set = 1, we obtain the mean of the EBE distribution. Variance, skewness and kurtosis measures can be easily derived from the well-known relationships. Three-dimensional plots of the skewness and kurtosis of the EBE model are presented in Figures 3 and 4.  These plots indicate that both measures depend very much on the shape parameter . The first four moments and the skewness and kurtosis of the EBE distribution for different values of parameters are represented in Table 1.  If we set = 1, we obtain the mean of the EBE distribution. Variance, skewness and kurtosis measures can be easily derived from the well-known relationships. Three-dimensional plots of the skewness and kurtosis of the EBE model are presented in Figures 3 and 4.  These plots indicate that both measures depend very much on the shape parameter . The first four moments and the skewness and kurtosis of the EBE distribution for different values of parameters are represented in Table 1.  These plots indicate that both measures depend very much on the shape parameter θ. The first four moments and the skewness and kurtosis of the EBE distribution for different values of parameters are represented in Table 1.

Proof. Starting with
finally, we get In the same way, the characteristic function of the EBE distribution becomes where i = √ −1 is the unit imaginary number.

Incomplete Moments
The s th lower and upper incomplete moments of Y are defined by where c (1) (∆,j) = θα 1 [α(1+ j)] ∆+κ+2 and γ(s, τ) = τ 0 y s−1 e −y dy is the lower incomplete gamma function. Similarly, the s th upper incomplete moment of the EBE distribution is where ζ(s, τ) = ∞ τ e −y y s−1 dy, is the upper incomplete gamma function.

Residual Life and Reversed Residual Life Functions
The r τh moment of the residual life via the general formula is given by (r,j) ζ(r + κ + 2, α(1 + j)τ) .
Using m 1,Y (τ) and m 2,Y (τ), one can obtain the "variance" and the "coefficient of variation" of the reversed residual life of the EBE distribution.

Estimation and Inference
Let Y 1 , Y 2 , . . . , Y n be a random sample of size n from EBE ψ . The log likelihood function for the vector of parameters λ, α and θ can be written as where m i = 1 + θαy i 2−θ and s i = e −αy i . The associated score function is given by The logL in (18) can be maximized by solving the nonlinear likelihood equations obtained by differentiating (18). The components of the score vector are given by

Simulation
The "inverse transform algorithm" is used to generate random data from the EBE distribution. We generated samples of sizes n = 50, 100, 200, 500 and 1000, and the simulations were repeated N = 1000 times from the EBE model for some parameter values. Tables 2 and 3 give the mean square errors (MSEs) and the biases, respectively. The average values of estimates (AVs), estimated average length (EAL) and the coverage probability (CP) are listed in Tables 4-6, respectively. From Table 2, we note that the AVs of estimates approach the initial values as n → ∞ , the MSEs for each parameter decrease to zero as n → ∞ , and the coverage lengths for each parameter decrease to zero as n → ∞. From Table 3, we note that the biases for each parameter are generally positive and decrease to zero as n → ∞ , and the coverage probabilities for each parameter approach the nominal level as n → ∞.

Modeling Stress-Rupture Life of Kevlar 49/Epoxy Strands Data
In this section, we illustrate the performance of the EBE distribution as compared to some alternative distributions using a real data application. The goodness-of-fit (GOF) statistics for this distribution are compared with other competitive distributions, and the maximum likelihood estimations (MLEs) of the distribution parameters are determined numerically. We compare the fits of the EBE distribution with the Burr type X (Burr X) distribution, Burr type XII (Burr XII) distribution, beta log logistic Weibull distribution (BLLW), beta Weibull log logistic (BWLL) and beta log logistic, beta linear failure rate geometric (BLFRG), exponentiated linear failure rate geometric (ELFRG), beta Rayleigh (BR), and beta Weibull geometric distributions (BWG) (see [23]). In order to compare the distributions, we consider the measures of GOF including the Akaike Information Criterion (C[1]), Bayesian Information Criterion (C [2]), Consistent Akaike Information Criterion (C [4]) and Hannan-Quinn Information Criterion (C [3]) statistics.
The following real data set represents the stress-rupture life of Kevlar 49/epoxy strands that are subjected to constant sustained pressure at the 90% stress level until all have failed that were provided by [24], given as 0.01, 0.08, 0.09, 0.09, 0. 10 Table 7 gives the MLE for all the models corresponds to the failure times data set. Table 8 shows the statistics for the failure times of the Kevlar data set. Figure 5 gives the kernel density estimation and box plot for exploring the symmetry of the stress-rupture life data. Figure 6 provides the fitted PDF in the left panel and fitted CDF in the right panel.   Based on Table 8, it is clear that the EBE distribution provides the best fit to these data, with −2logL = 143.3996, C 1 = 149.41, C 4 = 149.6887, C 2 = 152.3784 and C 3 = 152.3784. Thus, it is concluded that this model can be a better model than other competitive lifetime models for explaining the data set. Based on Figure 6, we note that the EBE distribution gives adequate fits. Many symmetric and near symmetric real-life data sets can be modeled using the new EBE model and found in [25][26][27][28][29][30][31][32][33][34][35][36]. For other right heavy tailed real data sets see [37][38][39][40][41][42][43][44][45]. As a future work we will consider "bivariate" and "multivariate" extensions of the EBE distribution. In particular with the "copula-based construction" method, "trivariate reduction" etc.

Conclusions
A new three-parameter lifetime distribution is proposed and studied. A simple-type Copulabased construction is presented to derive many bivariate-and multivariate-type distributions. We investigated some of mathematical properties such as the ordinary moments, moment generating function and conditional moment. Bonferroni and Lorenz curves and mean deviations are discussed. Residual life and reversed residual functions are also obtained. Some bivariate-and multivariate-   Based on Table 8, it is clear that the EBE distribution provides the best fit to these data, with −2logL = 143.3996, C 1 = 149.41, C 4 = 149.6887, C 2 = 152.3784 and C 3 = 152.3784. Thus, it is concluded that this model can be a better model than other competitive lifetime models for explaining the data set. Based on Figure 6, we note that the EBE distribution gives adequate fits. Many symmetric and near symmetric real-life data sets can be modeled using the new EBE model and found in [25][26][27][28][29][30][31][32][33][34][35][36]. For other right heavy tailed real data sets see [37][38][39][40][41][42][43][44][45]. As a future work we will consider "bivariate" and "multivariate" extensions of the EBE distribution. In particular with the "copula-based construction" method, "trivariate reduction" etc.

Conclusions
A new three-parameter lifetime distribution is proposed and studied. A simple-type Copulabased construction is presented to derive many bivariate-and multivariate-type distributions. We investigated some of mathematical properties such as the ordinary moments, moment generating function and conditional moment. Bonferroni and Lorenz curves and mean deviations are discussed. Residual life and reversed residual functions are also obtained. Some bivariate-and multivariate- Based on Table 8, it is clear that the EBE distribution provides the best fit to these data, with −2logL = 143.3996, C[1] = 149.41, C[4] = 149.6887, C[2] = 152.3784 and C [3] = 152.3784. Thus, it is concluded that this model can be a better model than other competitive lifetime models for explaining the data set. Based on Figure 6, we note that the EBE distribution gives adequate fits. Many symmetric and near symmetric real-life data sets can be modeled using the new EBE model and found in [25][26][27][28][29][30][31][32][33][34][35][36]. For other right heavy tailed real data sets see [37][38][39][40][41][42][43][44][45]. As a future work we will consider "bivariate" and "multivariate" extensions of the EBE distribution. In particular with the "copula-based construction" method, "trivariate reduction" etc.

Conclusions
A new three-parameter lifetime distribution is proposed and studied. A simple-type Copula-based construction is presented to derive many bivariate-and multivariate-type distributions. We investigated some of mathematical properties such as the ordinary moments, moment generating function and conditional moment. Bonferroni and Lorenz curves and mean deviations are discussed. Residual life and reversed residual functions are also obtained. Some bivariate-and multivariate-type extensions are proposed. The maximum likelihood method is used to estimate the model parameters. Finally, we illustrate the importance of the new model by studying real data applications to show the flexibility and potentiality of the new model. Author Contributions: All authors contributed equally to this work. All authors have read and agreed to the published version of the manuscript.
Funding: This research received no external funding.