A New Flexible Three-Parameter Model: Properties, Clayton Copula, and Modeling Real Data

: In this article, we introduced a new extension of the binomial-exponential 2 distribution. We discussed some of its structural mathematical properties. A simple type Copula-based construction is also presented to construct the bivariate- and multivariate-type distributions. We estimated the model parameters via the maximum likelihood method. Finally, we illustrated the importance of the new model by the study of two real data applications to show the ﬂexibility and potentiality of the new model in modeling skewed and symmetric data sets.


Introduction and Motivation
The monotonicity of the hazard (failure) rate function (HRF) of a life model plays an important role in modeling failure time data. Probability distributions with an increasing failure rate (IFR) have various applications in pricing and supply chain contracting studies. The IFR property is a well-known and useful concept in reliability theory, dynamic programming, and other areas of applied probability and statistics (See [1,2]). Recently, [3] introduced a new two-parameter lifetime model with IFR. The model of [3] is named the binomial-exponential2 (BE-2) model, which is constructed as the distribution of the random sum (RSum) of independent exponential random variables (IID RVs) when the sample size (n) has a zero truncated binomial (ZTB) model. The BE-2 distribution can be used as an alternative to the standard Weibull (W), standard gamma (Ga), exponentiated exponential (EE), and weighted exponential (WhE) distributions. The cumulative distribution function (CDF) of BE-2 distribution is given by: where α > 0 is the scale parameter, and β is the shape parameter, where 0 ≤ β ≤ 1. The probability density function (PDF) corresponding to (1) can be expressed as: The PDF in (1) can be written as: The survival function (SF) F γ (x) is given by: where γ = (1 − γ), for γ = 1, we get the baseline model, i.e.; F(x) = G(x), where the shape parameter γ is called the tilt parameter. The PDF corresponding to (4) can be expressed as: and the HRF is given by: The new PDF of the proposed lifetime model distribution can be right-skewed, symmetric, and left-skewed with many different useful shapes (see Figure 1), and this means that the new model will be suitable for modeling different real data sets, and the HRF of the new model exhibits many important HRF shapes such as the "increasing-constant", "decreasing", "increasing", "constant", and "bathtub" shapes (see Figure 2). Practically, the proposed lifetime model is much better than many competitive versions of the exponential model, such as the odd Lindley exponential, the Marshall-Olkin exponential, moment exponential, the logarithmic Burr-Hatke exponential, the generalized Marshall-Olkin exponential, beta exponential, the Marshall-Olkin-Kumaraswamy exponential, the Kumaraswamy exponential, and the Kumaraswamy-Marshall-Olkin exponential, so the new lifetime model may be a good alternative to these models in modeling relief times and survival times data sets.

Genesis of the New Model
In this section, we introduce the three parameters of the MOBE-2 distribution. Using (1) and (2) in Equations (4)−(6), we obtain the CDF, SF, and PDF of the MOBE-2 distribution, (for x > 0) with vector of parameters Ψ = (α, β, γ). The CDF and SF can be written as: respectively. The corresponding PDF can be derived as: Henceforth, let X ∼ MOBE-2(Ψ), with PDF (10). For the MOBE-2 distribution, the HRF can be written as: The MOBE-2 distribution is a very flexible model that approaches different distributions when its parameters are changed. For = 0, the MOBE-2 distribution reduces to the Marshall-Olkin extended exponential (MOEE) distribution. For = 1, we get MOEGa distribution with shape parameter 2 and scale parameter α. For = 1, we get BE-2 distribution (see Bakouch et al. (2014)). For β = 0 and = 1, we get the exponential (E) distribution. For β = γ = 1, the MOBE-2 distribution reduces to the Ga model with shape parameter 2 and scale parameter α. A useful representation for the new PDF is given in Appendix A. Figure 1 below gives some plots of the new PDF based on some selected parameters values. Based on Figure 1, we note that the new MOBE-2 distribution PDF can be right-skewed and left-skewed with many different useful shapes.
Symmetry 2020, xx, x FOR PEER REVIEW 3 of 20 respectively. The corresponding PDF can be derived as: Henceforth, let ∼ MOBE-2 , with PDF (10). For the MOBE-2 distribution, the HRF can be written as: The MOBE-2 distribution is a very flexible model that approaches different distributions when its parameters are changed. For = 0 , the MOBE-2 distribution reduces to the Marshall-Olkin extended exponential (MOEE) distribution. For = 1 , we get MOEGa distribution with shape parameter 2 and scale parameter . For = 1 , we get BE-2 distribution (see Bakouch et al. (2014)). For = 0 and = 1 , we get the exponential (E) distribution. For = = 1, the MOBE-2 distribution reduces to the Ga model with shape parameter 2 and scale parameter . A useful representation for the new PDF is given in Appendix A. Figure 1 below gives some plots of the new PDF based on some selected parameters values. Based on Figure 1, we note that the new MOBE-2 distribution PDF can be rightskewed and left-skewed with many different useful shapes.    From Figure 2 we note that the HRF of the new model exhibits many important HRF shapes, such as the increasing-constant (α = 1, β = 1, γ = 1), decreasing (α = 1, β = 0.5, γ = 0.1), increasing (α = 0.005, β = 1, γ = 1), constant (α = 0.001, β = 0.001, γ = 1), and bathtub (α = 0.05, β = 0.05, γ = 0.65) shapes.  The solution of the following relationship is used to find the quantile function (QF) of the MOBE-2 distribution, as follows: Since the uniform RVs are easily generated numerically in most statistical packages, the above scheme in (12) is very useful to generate MOBE-2 RVs and therefore can be easily implemented. It facilitates ready quantile-based statistical modeling. In particular, the median of is ( ), given by setting = in (12). Also using (12), we can determine the Bowley's skewness and the Moors' kurtosis. The Bowley's skewness is based on quartiles. Figure 3 indicates that both measures depend very much on the shape parameters . The solution of the following relationship is used to find the quantile function (QF) of the MOBE-2 distribution, as follows: Since the uniform RVs are easily generated numerically in most statistical packages, the above scheme in (12) is very useful to generate MOBE-2 RVs and therefore can be easily implemented. It facilitates ready quantile-based statistical modeling. In particular, the median of X is Q( 1 2 ), given by setting q = 1 2 in (12). Also using (12), we can determine the Bowley's skewness and the Moors' kurtosis. The Bowley's skewness is based on quartiles. Figure 3 indicates that both measures depend very much on the shape parameters β.
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Moments
, the ( ) moment of is given by: Where: Proof of Theorem 1. Let X be a RV following the MOBE-2 distribution. The r ( ) ordinary moment can be obtained using the well-known formula: Then: setting = (1 + ) , and after some algebra, the ( ) ordinary moment is given by: which complete the proof, where ( ) = refers to the gamma function. □

Moment Generating Function (MGF)
Theorem 2. If has the MOBE-2 ( ) , then the MGF of is given as follows:

Moments
where: Proof of Theorem 1. Let X be a RV following the MOBE-2 distribution. The r (th) ordinary moment can be obtained using the well-known formula: Then: x, and after some algebra, the r (th) ordinary moment is given by: which complete the proof, where Γ(n) = ∞

Moment Generating Function (MGF)
Theorem 2. If X has the MOBE-2 (Ψ), then the MGF of X is given as follows: where: Proof of Theorem 2. The MGF can be derived from: then we have: which can be written as: which completes the proof.

Conditional Moments
For any lifetime model, the s (th) lower (δ s (t)) and upper (π s (t)) IM of X is defined by: and respectively, for any real s > 0. The s (th) lower incomplete moment of MOBE-2 distribution is: where: Proof of Theorem 2. The MGF can be derived from: then we have: which can be written as: which completes the proof. □

Conditional Moments
For any lifetime model, the lower ( ) and upper ( ) IM of is defined by: and respectively, for any real 0. The lower incomplete moment of MOBE-2 distribution is: where: and: is the lower incomplete gamma function, where: , | , , ,...
the function 1 ⋅,⋅,⋅ is a called the confluent hypergeometric function. The first incomplete moment of , denoted by, , is computed using Equation (24) by setting 1 as: and: is the lower incomplete gamma function, where: the function 1F 1 [·, ·, ·] is a called the confluent hypergeometric function. The first incomplete moment of X, denoted by, δ 1 (t), is computed using Equation (24) by setting s = 1 as: where: Similarly, the s (th) upper incomplete moment of MOBE-2 distribution is: which can be written as: which completes the proof. □

Conditional Moments
For any lifetime model, the lower ( ) and upper ( ) IM of is defined by: and respectively, for any real 0. The lower incomplete moment of MOBE-2 distribution is: , where: and: is the lower incomplete gamma function, where: , | , , ,...
the function 1 ⋅,⋅,⋅ is a called the confluent hypergeometric function. The first incomplete moment of , denoted by, , is computed using Equation (24) by setting 1 as: where: tten as: the proof. □ oments ime model, the lower ( ) and upper ( ) IM of is defined by: ny real 0. The lower incomplete moment of MOBE-2 distribution is: , , , plete gamma function, where: , | , , ,...

Residual Life and Reversed Failure Rate Function
The n th order moment of the residual life is given by the general formula (see [5]): Applying the binomial expansion of (x − t) n and substituting f Ψ (x) given by (10) into the above formula gives: where ζ τ,κ = . The MRL of the MOBE-2 distribution is obtained by setting n = 1 in µ n (t). The variance of the residual life of the MOBE-2 distribution can be obtained easily by using µ 2 (t) and µ(t). The r-th moment of the reversed residual life (MRRL) can be obtained by the well-known formula: Applying the binomial expansion of (t − x) n and substituting f Ψ (x), given before, into the above formula gives: The mean waiting time (MWT) of the MOBE-2 distribution can be obtained by setting n = 1 in m n (t). Using m(t) and m 2 (t), one can obtain the variance and the coefficient of variation of the reversed residual life of the MOBE-2 distribution (for more details see [2]).

The Bivariate MOBE-2 Using the Morgenstern Family
First, we start with CDF for the Morgenstern family of two RVs (X 1 , X 2 ), which has the following form: setting: and: then we have a seven-dimension parameter model.

The Bivariate MOBE-2 Model
The bivariate extension via Clayton copula can be considered as a weighted version of the Clayton Copula, which is of the form This is indeed a valid copula. Next, let us assume that X ∼ MOBE-2 (γ 1 , α 1 , β 1 ) and Y ∼ MOBE-2 (γ 2 , α 2 , β 2 ). Then, setting: and: the associated CDF bivariate MOBE-2 type distribution will be: Note: depending on the specific baseline CDF, one may construct various bivariate MOBE-2 type models in which (δ 1 + δ 2 ) ≥ 0.

The Multivariate Extension
The d-dimensional version from the above will be: Further future works could be allocated for studying the bivariate and the multivariate extensions of the MOBE-2 model.

Estimation and Inference
Let X 1 , X 2 , . . . , X n be a random sample of size n from MOBE-2 (Ψ) where Ψ = (α, β, γ). The log likelihood function logL (Ψ) can be written as: The associated score function is given by U n (Ψ) = The log-likelihood can be maximized either directly or by solving the nonlinear likelihood equations obtained by differentiating logL (Ψ) . The components of the score vector are given by: and:

Concluding Remarks
In this paper, we introduced a new version of the BE-2 model. The new model is called MOBE-2 distribution. Some of its structural properties are also presented. A simple type Copula-based construction was also presented to construct the bivariate and the multivariate type distributions. We illustrated the importance of the new version by the study of two real data applications. The proposed

Concluding Remarks
In this paper, we introduced a new version of the BE-2 model. The new model is called MOBE-2 distribution. Some of its structural properties are also presented. A simple type Copula-based construction was also presented to construct the bivariate and the multivariate type distributions. We illustrated the importance of the new version by the study of two real data applications. The proposed lifetime model was much better than many competitive versions, such as the exponential, the odd Lindley exponential, the Marshall-Olkin exponential, moment exponential, the logarithmic Burr-Hatke exponential, the generalized Marshall-Olkin exponential, beta exponential, the Marshall-Olkin-Kumaraswamy exponential, the Kumaraswamy exponential, and the Kumaraswamy-Marshall-Olkin Exponential, so the new lifetime model may be a good alternative to these models in modeling relief times and survival times data sets.

Conflicts of Interest:
The authors declare no conflict of interest.