# A New Bivariate Extended Generalized Inverted Kumaraswamy Weibull Distribution

Authors: Mahmoud Ragab; Ahmed Elhassanein
Journal: Advances in Mathematical Physics
 (2022)

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## Abstract

This article presents a new bivariate extended generalized inverted Kumaraswamy Weibull (BIEGIKw-Weibull) distribution with nine parameters. Statistical properties of the new distribution are discussed. Forms of copulas, moments, conditional moments, bivariate reliability function, and bivariate hazard rate function are derived. Maximum likelihood estimators are formulated. Simulation is conducted for three different sets of parameters to verify the theoretical results and to discuss the new distribution properties. The performance of the maximum likelihood method is investigated via Monte Carlo simulation depending on the bias and the standard error. Simulated lifetime data is used as an application of the new model.


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## Body

## 1. Introduction

The inverse Weibull (IW) distribution is widely used because of its applicability in various fields, like medicine, statistics, engineering, physics, and fluid mechanics [1–11]. To enhance such distributions, researchers introduced new generators by supplementing shape parameters to the base line distribution. The inverted Kumaraswamy (IK) with two shape parameters has been derived by Abd AL-Fattah et al. [12]. To accommodate both monotonic and nonmonotonic failure rates, the IK distribution has been generalized to involve three shape parameters (GIKum) by Iqbal et al. [13]. A new version with five parameters (GIKw-W) has been introduced by Jamal et al. [14]. Although the univariate continuous models suit many types of data sets, they cannot be used to model dependent sets of data; therefore, a lot of efforts have been done to develop bivariate distributions. Muhammed [15] proposed a bivariate generalized Kumaraswamy distribution. A bivariate inverse Weibull distribution has been developed by Mondal and Kundu [16]. Darwish and Shahbaz [17] formulated a bivariate transmuted Burr distribution; see also [18–25]. Most of the developed bivariate distributions have different shapes for the joint pdf and have singular part. In some cases, their joint probability distribution function can be expressed in compact forms. The maximum likelihood estimators cannot be expressed in explicit forms in most of the cases. Ganji et al. [26] generalized the method introduced by Alzaatreh et al. [27] to generate bivariate distributions with marginals having T−X families. Let gX,Yx,y be the pdf of the bivariate random variable X,Y, with x∈a1,b1, y∈a2,b2,−∞<a1<b1<∞,−∞<a2<b2<∞. Consider F1GUu and F2GVv be functions of the cdfs of a random variables U and V, respectively, such that
(1)
F1GUu∈a1,b1 and F2GVv∈a2,b2(2)
F1GUu and F2GVv are differentiable and monotonically nondecreasing functions(3)
F1GUu⟶a1 as u⟶−∞, F1GUu⟶b1 as u⟶∞, F2GVv⟶a2 as v⟶−∞, and F2GVv⟶b2 as v⟶∞The cdf of the random variableU,V is given by
(1)FU,VF1GXu,F2GYz=∫c1F1GXu∫c2F2GYvgX,Yx,ydxdy.In this paper, we introduce a new bivariate extended generalized inverted Kumaraswamy Weibull (BIEGIKw-Weibull) distribution; its joint pdf is absolutely continuous, takes only one form with no singular parts, and offers different shapes for different values of parameters, and its hazard function shows different shapes. Almost all statistical quantities of the new distribution can be obtained in closed forms including the maximum likelihood estimators. The new model is developed using the new six parameter distribution that is more flexible with so favorable properties [28]. Theoretical properties of the proposed distribution including marginal distributions, copulas, moments, conditional moments, bivariate reliability function, and bivariate hazard function are computed. Theoretical properties are investigated via simulation. Monte Carlo simulation is used to discuss the goodness of fit and the availability of the maximum likelihood method. A real data application is presented that proves the applicability of the new distribution. The paper is organized as follows. The new distribution is formulated in Section 2. In Section 3, closed forms of moments are derived. Reliability and hazard function are computed in Section 4. Estimation is performed in Section 5. Simulation for different three sets of parameters is performed in Section 6. A real data application is discussed in Section 7. Conclusion is given in Section 8.


## 2. Model Description

A one-dimensional random variableZ is said to have a GIKw-Weibull distribution if its cumulative distribution function (cdf) is given by
(2)Fz=1−1+1−e−δzφ−λ−1−γ−αβ,where z>0,α,β,γ,δ,λ,φ>0 are shape parameters [29]. Using (2) as a baseline distribution and gX,Yx,y=1+δ11−x+δ21−y+2δ31−x−y in (1), where x,y∈0,1,δ1,δ2,δ3∈−1,1,−1≤δ1+δ3≤1, and −1≤δ2+δ3≤1, we formulate the following definition.Definition 1.
A bivariate random variableU,V is said to be a BIEGIKw-Weibull random variable if its cumulative probability function (cdf) and probability density function (pdf) are given by
(3)FU,Vu,v=1−1+1−e−δuφ−λ−1−γ−αβ×1−1+1−e−δvφ−λ−1−γ−αβ1+δ1+δ3×1−1−1+1−e−δuφ−λ−1−γ−αβ+δ2+δ3×1−1−1+1−e−δvφ−λ−1−γ−αβ,
fU,Vu,v=αβγλδφ2uvφ−1e−δuφ+vφ1−e−δuφ1−e−δvφ−λ+1×1−e−δuφ−λ−11−e−δvφ−λ−1−γ+1×1+1−e−δuφ−λ−1−γ1+1−e−δvφ−λ−1−γ−α+1×1−1+1−e−δuφ−λ−1−α×1−1+1−e−δvφ−λ−1−γ−αβ−1×1+δ1+δ31−21−1+1−e−δuφ−λ−1−γ−αβ+δ2+δ31−21−1+1−e−δvφ−λ−1−γ−αβ,
(4).where u,v>0,α,β,γ,δ>0,δ1,δ2,δ3∈−1,1,−1≤δ1+δ3≤1, and −1≤δ2+δ3≤1.


## 3. Marginals and Moments

Lemma 2.
LetU,V be a BIEGIKw-Weibull random variable with cdf and pdf given in (3) and (4). Then, the marginals are
(5)FUu=1−1+1−e−δuφ−λ−1−γ−αβ×1+δ1+δ31−1−1+1−e−δuφ−λ−1−γ−αβ,(6)FVv=1−1+1−e−δvφ−λ−1−γ−αβ×1+δ2+δ31−1−1+1−e−δvφ−λ−1−γ−αβ,(7)fUu=αβγλδφuφ−1e−δuφ1−e−δuφ−λ+11−e−δuφ−λ−1−γ+1×1+1−e−δuφ−λ−1−γ−α+1×1−1+1−e−δuφ−λ−1−γ−αβ−1×1+δ1+δ3−2δ1+δ31−1+1−e−δuφ−λ−1−γ−αβ,(8)fVv=αβγλδφvφ−1e−δvφ1−e−δvφ−λ+11−e−δvφ−λ−1−γ+1×1+1−e−δvφ−λ−1−γ−α+1×1−1+1−e−δvφ−λ−1−γ−αβ−1×1+δ2+δ3−2δ2+δ31−1+1−e−δvφ−λ−1−γ−αβ.Forδ1=δ2=−δ3, we get the baseline distribution EGIKw-Weibull.Copula function is commonly used to investigate the dependence between two random variables.Definition 3 (see [30]).
LetU,V be a BIGIKw-Weibull random variable with cpf FU,Vu,v and marginals FUu and FVv, and then, its copula function can be defied as FU,Vu,v≜Cυ,ϕ, where υ=FUu, ϕ=FVv, and υ,ϕ∈0,1and the copula density function is defined as cυ,ϕ=∂2Cυ,ϕ/∂υ∂ϕ.
Using chain rule, we obtain(9)fU,Vu,v=∂2Fu,v∂u∂v=∂2CFUu,FVv∂u∂v=∂2Cυ,ϕ∂υ∂ϕ∂FUu∂u∂FVv∂v≜cυ,ϕfUufVv.Lemma 4.
LetU,V be a BIEGIKw-Weibull random variable with pdf and marginals given in (4), (7), and (8). Then, the copula density function is given by
(10)cυ,ϕ=1Hu,δ1Hv,δ21+δ1+δ3×1−21−1+1−e−δuφ−λ−1−γ−αβ+δ2+δ31−21−1+1−e−δvφ−λ−1−γ−αβ.Lemma 5.
LetU,V be a BIEGIKw-Weibull random variable with pdf and marginals given in (4), (7), and (8). Then, the conditional density functions are
(11)fUVu/v=αβγλδφuφ−1e−δuφ1−e−δuφ−λ+1Hv,δ2×1−1−e−δuφλ−γ+11+1−e−δuφ−λ−1−γ−α+1×1−1+1−e−δuφ−λ−1−γ−αβ−1×1+δ1+δ31−21−1+1−e−δuφ−λ−1−γ−αβ+δ2+δ31−21−1+1−e−δvφ−λ−1−γ−αβ,(12)fVUv/u=αβγλδφvφ−1e−δvφ1−e−δvφ−λ+1Hu,δ11−1−e−δvφλ−γ−1×1+1−e−δvφ−λ−1−γ−α−1×1−1+1−e−δvφ−λ−1−γ−αβ−11+δ1+δ31−21−1+1−e−δuφ−λ−1−γ−αβ+δ2+δ31−21−1+1−e−δvφ−λ−1−γ−αβ,where
(13)Hx,δ=1+δ+δ33.7−2δ+δ31−21−1+1−e−δxφ−λ−1−γ−αβ.Lemma 6.
LetU,V be a BIEGIKw-Weibull random variable whose conditionals are given in (11) and (13). Then, the conditional moments are given by
(14)mUVrv=EUrV=v=1Hv,δ21δr/φ1+δ1+δ2+2δ3−2δ2+δ31−21−1+1−e−δvφ−λ−1−γ−αβ×∑i,j,k=0∞∑l=0λγj+k−1−1lξi,j,kλγj+k−1lΓr/φ+1l+1r/φ+1−2δ1+δ3δr/φ∑i,j,k=0∞∑l=02λγj+k−1−1lξi,j,kλγj+k−1l×Γr/φ+1l+1r/φ+1,mV/Uru=EVrU=u=1Hu,δ11δr/φ1+δ1+δ2+2δ3−2δ2+δ31−21−1+1−e−δuφ−λ−1−γ−αβ×∑i,j,k=0∞∑l=0λγj+k−1−1lξi,j,kβλγj+k−1lΓr/φ+1l+1r/φ+1−2δ1+δ3δr/φ∑i,j,k=0∞∑l=0λγj+k−1−1lξi,j,k2βλγj+k−1l×Γr/φ+1l+1r/φ+1,where
(15)ξi,j,kρ=−1i+jλγj+kρiαi+j−1jγj+k−1k.
γ is a positive integer and H is given by (13); for more details, see [28].Lemma 7.
LetU,V be a BIEGIKw-Weibull random variable whose density function is given in (4). Then, the joint moments are given by
(16)mr,s=EUrVs=1+δ1+δ2+2δ3δr/φδs/φ×∑i,j,k=0∞∑l=0λγj+k−1−1lξi,j,kβλγj+k−1lΓr/φ+1l+1r/φ+1×∑o,p,q=0∞∑t=0λγp+q−1−1tξi,j,kβλγp+q−1tΓs/φ+1t+1s/φ+1−2δr/φδs/φ∑i,j,k=0∞∑l=0λγj+k−1−1lξi,j,k2β2λγj+k−1lΓr/φ+1l+1r/φ+1×∑o,p,q=0∞∑t=0λγp+q−1−1tξo,p,qβλγp+q−1tΓs/φ+1t+1s/φ+1−2δ2+δ3δr/φδs/φ∑i,j,k=0∞∑l=0λγj+k−1−1lξi,j,kβλγj+k−1lΓr/φ+1l+1r/φ+1×∑o,p,q=0∞∑t=0λγj+q−1−1tξo,p,q2β2λγp+q−1tΓs/φ+1t+1s/φ+1,where γ is a positive integer and ξi,j,kρ is given by Equation (15).


## 4. Reliability and Hazard Functions

Bivariate hazard function can be used to characterize bivariate distributions. It describes the failure characteristics of the individual variables and their joint failure behavior. Here, we compute the bivariate reliability function and the hazard function defined by Navarro [30].Lemma 8.
LetU,V be a BIEGIKw-Weibull random variable whose cumulative and marginals are given in (3), (5), and (6), respectively. Then, its bivariate reliability function is given by
(17)Ru,v=R1u,v+R2u,v,where
(18)R1u,v=1−1−1+1−e−δuφ−λ−1−γ−αβ×1+δ1+δ31−1+1−e−δuφ−λ−1−γ−αβ−1−1+1−e−δvφ−λ−1−γ−αβ×1+δ2+δ31−1+1−e−δvφ−λ−1−γ−αβ,R2u,v=αβγλδφ2uvφ−1e−δuφ+vφ1−e−δuφ1−e−δvφ−λ+1×1−e−δuφ−λ−11−e−δvφ−λ−1−γ+1×1+1−e−δuφ−λ−1−γ1+1−e−δvφ−λ−1−γ−α+1×1−1+1−e−δuφ−λ−1−γ−α×1−1+1−e−δvφ−λ−1−γ−αβ−1×1+δ1+δ31−21−1+1−e−δuφ−λ−1−γ−αβ+δ2+δ31−21−1+1−e−δvφ−λ−1−γ−αβ.Lemma 9.
LetU,V be a BIEGIKw-Weibull random variable whose density and reliability functions are given in (4) and (17). Then, its bivariate hazard rate function is given by
(19)hu,v=αβγλδφ2uvφ−1e−δuφ+vφ1−e−δuφ1−e−δvφ−λ+1×1−e−δuφ−λ−11−e−δvφ−λ−1−γ+1×1+1−e−δuφ−λ−1−γ1+1−e−δvφ−λ−1−γ−α+1×1−1+1−e−δuφ−λ−1−γ−α×1−1+1−e−δvφ−λ−1−γ−αβ−1×1+δ1+δ31−21−1+1−e−δuφ−λ−1−γ−αβ+δ2+δ31−21−1+1−e−δvφ−λ−1−γ−αβ1−1−1+1−e−δuφ−λ−1−γ−αβ×1+δ1+δ31−1+1−e−δuφ−λ−1−γ−αβ−1−1+1−e−δvφ−λ−1−γ−αβ×1+δ2+δ31−1+1−e−δvφ−λ−1−γ−αβ+αβγλδφ2uvφ−1e−δuφ+vφ1−e−δuφ1−e−δvφ−λ+1×1−e−δuφ−λ−11−e−δvφ−λ−1−γ+1×1+1−e−δuφ−λ−1−γ1+1−e−δvφ−λ−1−γ−α+1×1−1+1−e−δuφ−λ−1−γ−α×1−1+1−e−δvφ−λ−1−γ−αβ−1×1+δ1+δ31−21−1+1−e−δuφ−λ−1−γ−αβ+δ2+δ31−21−1+1−e−δvφ−λ−1−γ−αβ−1.


## 5. Estimation

The maximum likelihood method is used to perform point estimators of the unknown parametersΘ=α,β,γ,λ,φ,δ,δ1,δ2,δ3′. Let u1,v1,u2,v2,⋯,un,vn be a random sample from the BIEGIKw-Weibull random variable. The maximum log-likelihood function is given by LΘ=L1Θ+L2Θ, where
(20)L1Θ=2nlnαβγλδφ+φ−1∑i=1nlnui−δ∑i=1nuiφ−λ+1×∑i=1nln1−e−δuiφ−γ+1∑i=1nln1−e−δuiφ−λ−1−α+1×∑i=1nln1+1−e−δuiφ−λ−1−γ+β−1×∑i=1nln1−1+1−e−δuiφ−λ−1−γ−α+,L2Θ=φ−1∑i=1nlnvi−δ∑i=1nviφ−λ+1∑i=1nln1−e−δviφ−γ+1×∑i=1nln1−e−δviφ−λ−1−α+1∑i=1nln1+1−e−δviφ−λ−1−γ+β−1∑i=1nln1−1+1−e−δviφ−λ−1−γ−α+∑i=1nln1+δ1+δ31−21−1+1−e−δuiφ−λ−1−γ−αβ+δ2+δ31−21−1+1−e−δviφ−λ−1−γ−αβ.Consider the score vectorΨ=Ψα,Ψβ,Ψγ,Ψλ,Ψφ,Ψδ,Ψδ1,Ψδ2,Ψδ3′, where Ψα=Ψα1+Ψα2,Ψβ=Ψβ1+Ψβ2,Ψγ=Ψγ1+Ψγ2,Ψλ=Ψλ1+Ψλ2,Ψφ=Ψφ1+Ψφ2+Ψφ3,Ψδ=Ψδ1+Ψδ2; then,
(21)Ψα1=2nα−∑i=1nln1+1−e−δuiϕ−λ−1−γ−∑i=1nln1+1−e−δviϕ−λ−1−γ+β−1×∑i=1n1+1−e−δuiϕ−λ−1−γ−a1−1+1−e−δuiϕ−λ−1−γ−a+∑i=1n1+1−e−δviϕ−λ−1−γ−a1−1+1−e−δviϕ−λ−1−γ−a,Ψα2=−2δ1+δ3β∑i=1n1−1+1−e−δuiϕ−λ−1−γ−αβ−1Hui,vi×1+1−e−δuiϕ−λ−1−γ−aln1+1−e−δuiϕ−λ−1−γ−2δ2+δ3β∑i=1n1−1+1−e−δviϕ−λ−1−γ−αβ−1Hui,vi×1+1−e−δviϕ−λ−1−γ−aln1+1−e−δviϕ−λ−1−γ,Ψβ1=2nβ+∑i=1nln1−1+1−e−δuiϕ−λ−1−γ−a+∑i=1nln1−1+1−e−δviϕ−λ−1−γ−α,Ψβ2=−2δ1+δ3∑i=1n1−1+1−e−δuiϕ−λ−1−γ−αβHui,vi×ln1−1+1−e−δuiϕ−λ−1−γ−a−2δ2+δ3∑i=1n1−1+1−e−δviϕ−λ−1−γ−aβHui,vi×ln1−1+1−e−δviϕ−λ−1−γ−a,Ψγ1=2nγ−∑i=1nln1−e−δuiφ−λ−1−∑i=1nln1−e−δviφ−λ−1+α+1×∑i=1n1−e−δuiφ−λ−1−γln1−e−δuiφ−λ−1+α+1×∑i=1n1−e−δuiφ−λ−1−γln1−e−δuiφ−λ−1−β−1α×∑i=1n1+1−e−δuiφ−λ−1−γ−α+11−1+1−e−δuiφ−λ−1−γ−α1−e−δuiφ−λ−1−γ×ln1−e−δuiφ−λ−1+∑i=1n1+1−e−δviφ−λ−1−γ−α+11−1+1−e−δviφ−λ−1−γ−α×1−e−δviφ−λ−1−γln1−e−δviφ−λ−1,Ψγ2=2δ1+δ3αβ∑i=1n1−1+1−e−δuiφ−λ−1−γ−αβ−1Hui,vi×1+1−e−δuiφ−λ−1−γ−α+11−e−δuiφ−λ−1−γ×ln1−e−δuiφ−λ−1+2δ2+δ3αβ×∑i=1n1−1+1−e−δviφ−λ−1−γ−αβ−1Hui,vi×1+1−e−δviφ−λ−1−γ−α+11−e−δviφ−λ−1−γ×ln1−e−δviφ−λ−1,Ψλ1=2nλ−∑i=1nln1−e−δuiφ−λ−1−∑i=1nln1−e−δviφ−λ−1+α+1×∑i=1n1−e−δuiφ−λ−1−γln1−e−δuiφ−λ−1+α+1×∑i=1n1−e−δviφ−λ−1−γln1−e−δviφ−λ−1−β−1α×∑i=1n1+1−e−δuiφ−λ−1−γ−α+11−1+1−e−δuiφ−λ−1−γ−α1−e−δuiφ−λ−1−γ×ln1−e−δuiφ−λ−1−γ+∑i=1n1+1−e−δviφ−λ−1−γ−α+11−1+1−e−δviφ−λ−1−γ−α×1−e−δviφ−λ−1−γln1−e−δviφ−λ−1,Ψλ2=2δ1+δ3αβ∑i=1n1−1+1−e−δuiφ−λ−1−γ−αβ−1Hui,vi×1+1−e−δuiφ−λ−1−γ−α+11−e−δuiφ−λ−1−γ×ln1−e−δuiφ−λ−1+2δ2+δ3αβ×∑i=1n1−1+1−e−δviφ−λ−1−γ−αβ−1Hui,vi×1+1−e−δviφ−λ−1−γ−α+11−e−δviφ−λ−1−γ×ln1−e−δviφ−λ−1,Ψϕ1=2nϕ+∑i=1nlnui+lnvi−δ∑i=1nuiϕlnui+viϕlnvi−λ+1δ×∑i=1nuiϕe−δuiϕlnui1−e−δuiϕ+viϕe−δviϕlnvi1−e−δviϕ+γ+1λδ,∑i=1nuiϕ1−e−δuiϕ−λ+1e−δuiϕlnui1−e−δuiϕ−λ−1+viϕ1−e−δviϕ−λ+1e−δviϕlnvi1−e−δviϕ−λ−1−α+1γλδ∑i=1nuiϕ1−e−δuiϕ−λ+1e−δuiϕlnui1+1−e−δuiϕ−λ−1−γ×1−e−δuiϕ−λ−1−γ+1+viϕ1−e−δviϕ−λ+1e−δviϕlnvi1+1−e−δviϕ−λ−1−γ×1−e−δviϕ−λ−1−γ+1,Ψϕ2=β−1αγλδ∑i=1nuiϕ1−e−δuiϕ−λ+1e−δuiϕlnui1−1+1−e−δuiϕ−λ−1−γ−α×1−e−δuiϕ−λ−1−γ+11+1−e−δuiϕ−λ−1−γ−α+1+viϕ1−e−δviϕ−λ+1e−δviϕlnvi1−1+1−e−δviϕ−λ−1−γ−α1−e−δviφ−λ−1−γ+1×1+1−e−δviφ−λ−1−γ−α+1−2δ1+δ3αβγλδ,∑i=1nuiϕ1−1+1−e−δuiφ−λ−1−γ−αβ−1Hui,vi×1+1−e−δuiφ−λ−1−γ−α+1e−δuiϕlnui×1−e−δuiφ−λ−1−γ+11−e−δuiφ−λ+1,Ψϕ3=−2δ2+δ3αβγλδ∑i=1nviϕ1−1+1−e−δviφ−λ−1−γ−αβ−1Hui,vi×1+1−e−δviφ−λ−1−γ−α+1e−δviϕlnvi×1−e−δviφ−λ−1−γ+11−e−δviφ−λ+1,Ψδ1=2nδ−∑i=1nuiϕ+viϕ+λ+1∑i=1nuiϕe−δuiϕ1−e−δuiϕ+viϕe−δviϕ1−e−δviϕ+γ+1λ∑i=1nuiϕe−δuiϕ1−e−δuiϕ−λ+11−e−δuiϕ−λ−1+viϕe−δviϕ1−e−δviϕ−λ+11−e−δviϕ−λ−1,α+1γλ∑i=1nuiϕe−δuiϕ1−e−δuiϕ−λ+11+1−e−δuiϕ−λ−1−γ1−e−δuiϕ−λ−1−γ+1+viϕe−δviϕ1−e−δviϕ−λ+11+1−e−δviϕ−λ−1−γ1−e−δviφ−λ−1−γ+1,Ψδ2=β−1αγλ∑i=1nuiϕe−δuiϕ1−e−δuiϕ−λ+11−1+1−e−δuiϕ−λ−1−γ−α×1−e−δuiφ−λ−1−γ+11+1−e−δuiφ−λ−1−γ−α+1+viϕe−δviϕ1−e−δviϕ−λ+11−1+1−e−δviϕ−λ−1−γ−α×1−e−δviφ−λ−1−γ+11+1−e−δviφ−λ−1−γ−α+1−2δ1+δ3αβγλ×∑i=1nuiϕ1−1+1−e−δuiφ−λ−1−γ−αβ−1Hui,vi×1+1−e−δuiφ−λ−1−γ−α+1e−δuiϕ×1+1−e−δuiφ−λ−1−γ−γ+11−e−δuiφ−λ+1−2δ2+δ3αβγλδ∑i=1nviϕ1−1+1−e−δviφ−λ−1−γ−αβ−1Hui,vi×1+1−e−δviφ−λ−1−γ−α+1e−δviϕ×1−e−δviφ−λ−1−γ+11−e−δviφ−λ+1,Ψδ1=n−2∑i=1n1−1+1−e−δuiφ−λ−1−γ−αβHui,vi,Ψδ2=n−2∑i=1n1−1+1−e−δviφ−λ−1−γ−αβHui,vi,Ψδ3=2n−2∑i=1n1−1+1−e−δuiφ−λ−1−γ−αβHui,vi+1−1+1−e−δviφ−λ−1−γ−αβHui,vi,where
(22)Hui,vi=1+δ1+δ31−21−1+1−e−δuiφ−λ−1−γ−αβ+δ2+δ31−21−1+1−e−δviφ−λ−1−γ−αβ.The system of equations can be solved via symbolic mathematical packages.


## 6. Simulation

In this section, simulation is performed for three different sets of parameters to verify our theoretical results and to discuss the new bivariate distribution properties. For the first set of parameters,α=1, β=1.8,γ=2.0,λ=1.5,δ=1.6,δ1=−0.6,δ2=−0.5, δ3=0.7, and ϕ=2.5. The symmetry, unimodality, and high beakness of the density function (4) can be observed in Figure 1(a). For u,v approximately greater than 2,2, the cumulative function (3) approaches 1 (Figure 1(b)). The bivariate hazard function increases with the increase of u,v as displayed in Figure 1(c). Symmetry of marginals and bell shape densities with beakness can be noted in Figures 1(d)–1(g). For the second set of parameters, α=1.5, β=1.0,γ=0.7,λ=2.0,δ=0.6,δ1=0.7,δ2=0.8, δ3=−0.5, and ϕ=1.0. The density function (4) is unimodal and has high beak (Figure 2(a)). For u,v approximately greater than 10,10, the cumulative function (3) approaches 1 (Figure 2(b)). The bivariate hazard function decreases with the increase of u,v (Figure 2(c)). Unimodality and right tail property can be observed for marginals (Figures 2(d)–2(g)). For the third set of parameters, α=0.5, β=0.7,γ=0.9,λ=0.6,δ=0.2,δ1=0.5,δ2=0.4, δ3=0.1, and ϕ=0.3. The density function (4) deformed and becomes an open surface (Figure 3(a)). The cumulative function (3) approaches 1 as u,v approaches infinity (Figure 3(b)). The bivariate hazard function decreases with the increase of u,v (Figure 3(c)). Marginals are given in Figures 3(d)–3(g). Monte Carlo simulation is used to generate samples for different sizes to discuss the performance of the new distribution. The first set of observations is generated for the parameters α=0.5, β=3,γ=0.5,λ=0.6,δ=1.5,δ1=−0.7,δ2=−0.6, δ3=0.9, and ϕ=3; the second set is generated for the parameters α=1.7, β=1.8,γ=1.0,λ=1.2,δ=1.6,δ1=−0.6,δ2=−0.5, δ3=0.8, and ϕ=0.7. Considering δ1, δ2, and δ3 as chosen parameters, the maximum likelihood method is used to estimate the rest parameters; results are given in Tables 1 and 2. Depending on the bias and standard error, the maximum likelihood method shows well performance.Figure 1
For the set of parametersα=1, β=1.8,γ=2.0,λ=1.5,δ=1.6,δ1=−0.6,δ2=−0.5, δ3=0.7, and ϕ=2.5, the density function (4) is given in (a), the cumulative function (3) is given in (b), the hazard function (17) is given in (c), and the marginals (5), (7), (6), and (8) are given in (d–g), respectively.
(a)(b)(c)(d)(e)(f)(g)Figure 2
For the set of parametersα=1.5,β=1.0,γ=0.7,λ=2.0,δ=0.6,δ1=0.7,δ2=0.8, δ3=−0.5, and ϕ=1.0, the density function (4) is given in (a), the cumulative function (3) is given in (b), the hazard function (17) is given in (c), and the marginals (5), (7), (6), and (8) are given in (d–g), respectively.
(a)(b)(c)(d)(e)(f)(g)Figure 3
For the set of parametersα=0.5,β=0.7,γ=0.9,λ=0.6,δ=0.2,δ1=0.5,δ2=0.4, δ3=0.1, and ϕ=0.3, the density function (4) is given in (a), the cumulative function (3) is given in (b), the hazard function (17) is given in (c), and the marginals (5), (7), (6), and (8) are given in (d–g), respectively.
(a)(b)(c)(d)(e)(f)(g)Table 1
Average maximum likelihood estimates for the first set of observations.
Sample sizeParameterAverage estimateBiasSEn=20α0.3420-0.1580.0279β2.510-0.490.134γ0.5350.0350.102λ0.5730.0730.118δ1.216-0.2840.143φ2.902-0.0980.103n=50α0.4829-0.0170.0914β4.5081.5080.667γ0.54220.0420.0986λ0.4006-0.0990.0773δ0.858-0.6420.153φ2.801-0.1990.177n=100α0.2661-0.2440.0203β3.2060.2060.243γ0.91530.41530.0441λ0.4187-0.08130.0748δ0.3047-0.19530.0325φ3.15960.15960.0850Table 2
Average maximum likelihood estimates for the second set of observations.
Sample sizeParameterAverage estimateBiasSEn=20α1.463-0.2370.185β1.980.181.40γ0.96-0.041.24λ1.1480.0520.366δ1.9910.3910.561φ0.8830.1830.240n=50α1.405-0.2590.104β2.4240.6240.956γ1.390.391.21λ1.4970.2970.579δ1.9410.3410.378φ0.714-0.0140.310n=100α1.448-0.2510.0804β1.363-0.4370.332γ1.080.081.76λ1.4030.2030.309δ1.323-0.1230.422φ0.63-0.070.276


## 7. Real Data Application

In this section, the data discussed by [31] is used to investigate the applicability of the new distribution. Let u1,v1,u2,v2,⋯,u50,v50 be observed values of a BIEGIKw-Weibull random variable U,V with parameters α,β,γ,λ,δ,ϕ,δ1,δ2,δ3, where U is the processor lifetime and V is the memory lifetime; see Table 3 and Figure 4. The maximum likelihood method is conducted to estimate the parameters with AIC and BIC; see Table 4. The joint density function, hazard function, and marginals for the estimated parameters are given in Figure 5.Table 3
Simulated computer series system of two components.
SysProcessor lifetimeMemory lifetimeSysProcessor lifetimeMemory lifetimeSysProcessor lifetimeMemory lifetimeSysProcessor lifetimeMemory lifetimeSysProcessor lifetimeMemory lifetime11.92923.9291111.93864.0043211.17393.3857310.11810.0884410.62701.728923.66210.0026122.10002.0513221.34821.9705325.05332.3238420.79470.794733.96080.8323130.98670.9867233.09353.0935331.64652.0197430.50795.353542.35043.3364140.18370.1837242.13962.1548340.90960.6214442.59132.591351.08333.3059151.39894.1268251.32880.9689351.74942.3643452.53722.492362.84141.8438162.37572.7953260.11150.1115360.10580.1058461.19170.080170.33090.3309173.52021.4095270.85032.8578370.45930.4593471.52544.408882.98841.5961182.33640.1624280.19550.1955380.99381.7689481.09861.098690.57841.8795190.85841.9556290.46140.8584395.75610.3212491.00511.0051100.55200.5520204.34351.0001303.38871.9796406.29501.0495501.36401.3640Figure 4
Histogram of the data in Table3.
(a)(b)Table 4
The average maximum likelihood estimates of the BIEGIKw-Weibull parameters for the data in Table3.
ParameterEstimateAICBICα0.82096924β0.38096869γ1.03547483λ1.25168661δ0.23330064325.4836342.6918ϕ1.65658910δ1-0.36575551δ20.86741494δ3-0.05319925Figure 5
(a) The joint density function, (b) the cumulative function, (c) the hazard function, and (d–g) the marginals for the estimated parameters in Table4.
(a)(b)(c)(d)(e)(f)(g)


## 8. Conclusions

Analysis of correlated data is one of the most important problems in statistics and data science. Here, we introduced a new bivariate distribution named BIEGIKw-Weibull. The proposed model is with nine parameters. It is a flexible one. Theoretical properties including density function, cumulative function, marginals, copula function, conditional distributions, and conditional moments have been derived explicitly. The new model exhibits very rich characteristics that differ according to the parameters. That supports the applicability of the model for a large set of correlated data with various properties. Simulation clearly verified the theoretical properties and the richness of its preferable characteristics. For different values of the parameters, the distribution has extremely different properties that are clear from Figures1–3. For a set of values, we can observe symmetry and close surface for density function side by side with unimodality (Figure 1). All these properties deformed for another set of parameters (Figure 3). The bivariate hazard function exhibited different shapes. Monte Carlo simulation and real data application prove the applicability of the new distribution and the availability of the maximum likelihood method.



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