The bivariate Weibull distribution based on the GFGM copula

M. A. Abd Elgawad; H. M. Barakat; Atef F. Hashem; G. M. Mansour; I. A. Husseiny; M. A. Alawady; M. O. Mohamed; M. A. Abd Elgawad; H. M. Barakat; Atef F. Hashem; G. M. Mansour; I. A. Husseiny; M. A. Alawady; M. O. Mohamed

doi:10.3934/math.20251224

AIMS Mathematics

2025, Volume 10, Issue 11: 27862-27897. doi: 10.3934/math.20251224

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The bivariate Weibull distribution based on the GFGM copula

1.
Department of Mathematics and Statistics, Faculty of Science, Imam Mohammad Ibn Saud Islamic University (IMSIU), Riyadh 11432, Saudi Arabia
2.
Department of Mathematics, Faculty of Science, Zagazig University, Zagazig 44519, Egypt
3.
Department of Statistics and Operation Research, College of Science, Qassim University, Buraydah 51482, Saudi Arabia
4.
Faculty of Computers and Information Systems, Egyptian Chinese University, Cairo 11528, Egypt

Received: 23 July 2025 Revised: 05 November 2025 Accepted: 17 November 2025 Published: 28 November 2025
MSC : 60B12, 62G30

In statistical modeling, bivariate models are essential, especially when examining data with two associated variables. Bivariate distributions capture dependencies between variables, offering a more realistic depiction of real-world phenomena compared to univariate models that treat variables independently. This is particularly important in domains where variables frequently show non-trivial correlations, such as environmental science, reliability engineering, medicine, and finance. This motivates the proposal of a bivariate distribution that uses the generalized Farlie-Gumbel-Morgenstern (FGM) copula and Weibull marginal distribution, referred to as the GFGM-WD. The GFGM-WD describes bivariate lifetime data with weak to moderate correlation between variables. The suggested model was employed to investigate the reliability of dependent stress-strength models. Several properties of the GFGM-WD were derived, including the product moment, the coefficient of correlation between the inner variables, and the conditional expectation. Additionally, the statistical characteristics of the concomitants' k-record values from the GFGM-WD were discussed. We ran comprehensive Monte Carlo simulations to assess the suggested distribution's performance and used the maximum likelihood estimation and Bayesian methods to estimate its parameters. Finally, the distribution was tested on two actual medical datasets, showing that it outperformed other pre-existing bivariate models in terms of fitting accuracy.
- generalized FGM copula,
- stress-strength model,
- Bayesian estimation,
- entropy,
- extropy,
- bivariate data analysis,
- Weibull distribution,
- concomitants,
- k-record values
Citation: M. A. Abd Elgawad, H. M. Barakat, Atef F. Hashem, G. M. Mansour, I. A. Husseiny, M. A. Alawady, M. O. Mohamed. The bivariate Weibull distribution based on the GFGM copula[J]. AIMS Mathematics, 2025, 10(11): 27862-27897. doi: 10.3934/math.20251224

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Abstract

In statistical modeling, bivariate models are essential, especially when examining data with two associated variables. Bivariate distributions capture dependencies between variables, offering a more realistic depiction of real-world phenomena compared to univariate models that treat variables independently. This is particularly important in domains where variables frequently show non-trivial correlations, such as environmental science, reliability engineering, medicine, and finance. This motivates the proposal of a bivariate distribution that uses the generalized Farlie-Gumbel-Morgenstern (FGM) copula and Weibull marginal distribution, referred to as the GFGM-WD. The GFGM-WD describes bivariate lifetime data with weak to moderate correlation between variables. The suggested model was employed to investigate the reliability of dependent stress-strength models. Several properties of the GFGM-WD were derived, including the product moment, the coefficient of correlation between the inner variables, and the conditional expectation. Additionally, the statistical characteristics of the concomitants' k-record values from the GFGM-WD were discussed. We ran comprehensive Monte Carlo simulations to assess the suggested distribution's performance and used the maximum likelihood estimation and Bayesian methods to estimate its parameters. Finally, the distribution was tested on two actual medical datasets, showing that it outperformed other pre-existing bivariate models in terms of fitting accuracy.

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