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.
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 families. Let be the pdf of the bivariate random variable , with , . Consider and be functions of the cdfs of a random variables and , respectively, such that (1) and (2) and are differentiable and monotonically nondecreasing functions(3) as , as , as , and as
The cdf of the random variable 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.
2. Model Description
A one-dimensional random variable is said to have a GIKw-Weibull distribution if its cumulative distribution function (cdf) is given by where are shape parameters [29]. Using (2) as a baseline distribution and in (1), where , and , we formulate the following definition.
Definition 1. A bivariate random variable is said to be a BIEGIKw-Weibull random variable if its cumulative probability function (cdf) and probability density function (pdf) are given by , where , and
3. Marginals and Moments
Lemma 2. Let be a BIEGIKw-Weibull random variable with cdf and pdf given in (3) and (4). Then, the marginals are
For , we get the baseline distribution EGIKw-Weibull.
Copula function is commonly used to investigate the dependence between two random variables.
Definition 3 (see [30]). Let be a BIGIKw-Weibull random variable with cpf and marginals and , and then, its copula function can be defied as , where , , and and the copula density function is defined as
Using chain rule, we obtain
Lemma 4. Let be a BIEGIKw-Weibull random variable with pdf and marginals given in (4), (7), and (8). Then, the copula density function is given by
Lemma 5. Let be a BIEGIKw-Weibull random variable with pdf and marginals given in (4), (7), and (8). Then, the conditional density functions are where
Lemma 6. Let be a BIEGIKw-Weibull random variable whose conditionals are given in (11) and (13). Then, the conditional moments are given by where is a positive integer and is given by (13); for more details, see [28].
Lemma 7. Let be a BIEGIKw-Weibull random variable whose density function is given in (4). Then, the joint moments are given by where is a positive integer and 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. Let 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 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
5. Estimation
The maximum likelihood method is used to perform point estimators of the unknown parameters . Let be a random sample from the BIEGIKw-Weibull random variable. The maximum log-likelihood function is given by , where
Consider the score vector , where ; then, where
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, , , , and . The symmetry, unimodality, and high beakness of the density function (4) can be observed in Figure 1(a). For approximately greater than , the cumulative function (3) approaches 1 (Figure 1(b)). The bivariate hazard function increases with the increase of 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, , , , and . The density function (4) is unimodal and has high beak (Figure 2(a)). For approximately greater than , the cumulative function (3) approaches 1 (Figure 2(b)). The bivariate hazard function decreases with the increase of (Figure 2(c)). Unimodality and right tail property can be observed for marginals (Figures 2(d)–2(g)). For the third set of parameters, , , , and . The density function (4) deformed and becomes an open surface (Figure 3(a)). The cumulative function (3) approaches 1 as approaches infinity (Figure 3(b)). The bivariate hazard function decreases with the increase of (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 , , , and ; the second set is generated for the parameters , , , and . Considering , , and 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.
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7. Real Data Application
In this section, the data discussed by [31] is used to investigate the applicability of the new distribution. Let be observed values of a BIEGIKw-Weibull random variable with parameters , where is the processor lifetime and 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.
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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 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.
Acknowledgments
This project was funded by the Deanship of Scientific Research (DSR) at King Abdulaziz University (KAU), Jeddah, Saudi Arabia, under grant no. G: 80-351-1442. The authors therefore acknowledge with thanks DSR for the technical and financial support.