A New Bivariate Extended Generalized Inverted Kumaraswamy Weibull Distribution

Information Technology Department, Faculty of Computing and Information Technology, King Abdulaziz University, Jeddah 21589, Saudi Arabia Mathematics Department, Faculty of Science, Al-Azhar University, Naser City, 11884 Cairo, Egypt Centre of Artificial Intelligence for Precision Medicines, King Abdulaziz University, Jeddah 21589, Saudi Arabia Department of Mathematics, College of Science, University of Bisha, Bisha, Saudi Arabia Department of Mathematics, Faculty of Science, Damanhour University, Damanhour, Egypt


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][2][3][4][5][6][7][8][9][10][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 Kumaras-wamy 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][19][20][21][22][23][24][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 g X,Y ðx, yÞ be the pdf of the bivariate random variable ðX, YÞ, with x ∈ a 1 , b 1 , y ∈ a 2 , b 2 , −∞ < a 1 < b 1 <∞ ,−∞<a 2 < b 2 <∞. Consider F 1 ðG U ðuÞÞ and F 2 ðG V ðvÞÞ be functions of the cdfs of a random variables U and V, respectively, such that (1) F 1 ðG U ðuÞÞ ∈ ½a 1 , b 1 and F 2 ðG V ðvÞÞ ∈ ½a 2 , b 2 (2) F 1 ðG U ðuÞÞ and F 2 ðG V ðvÞÞ are differentiable and monotonically nondecreasing functions The cdf of the random variable ðU, VÞ is given by 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.

Marginals and Moments
Lemma 2. Let ðU, VÞ be a BIEGIKw-Weibull random variable with cdf and pdf given in (3) and (4). Then, the marginals are For δ 1 = δ 2 = −δ 3 , we get the baseline distribution EGIKw-Weibull. Copula function is commonly used to investigate the dependence between two random variables.

Lemma 7.
Let ðU, VÞ be a BIEGIKw-Weibull random variable whose density function is given in (4). Then, the joint moments are given by where γ is a positive integer and ξ i,j,k ðρÞ is given by Equation (15).

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. Let ðU, VÞ 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 where Lemma 9. Let ðU, VÞ 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 Advances in Mathematical Physics

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Advances in Mathematical Physics 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.

Real Data Application
In this section, the data discussed by [31] is used to investigate the applicability of the new distribution. Let ðu 1 , v 1 Þ, ðu 2 , v 2 Þ, ⋯, ðu 50 , v 50 Þ be observed values of a BIEGIKw-Weibull random variable ðU, VÞ with parameters ðα, β, γ, λ, δ, ϕ, δ 1 , δ 2 , δ 3 Þ, 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.

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 Figures 1-3. 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.

Data Availability
The data used are included within the article.

Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this article.  Table 4. 12 Advances in Mathematical Physics