Bivariate Chen distribution based on iterated FGM copula: Properties and applications

M. A. Alawady; Mohamed A. Abd Elgawad; Salem A. Alyami; H. M. Barakat; I. A. Husseiny; G. M. Mansour; T. S. Taher; M. O. Mohamed; M. A. Alawady; Mohamed A. Abd Elgawad; Salem A. Alyami; H. M. Barakat; I. A. Husseiny; G. M. Mansour; T. S. Taher; M. O. Mohamed

doi:10.3934/math.2026222

AIMS Mathematics

2026, Volume 11, Issue 3: 5379-5408. doi: 10.3934/math.2026222

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Research article

Bivariate Chen distribution based on iterated FGM copula: Properties and applications

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

Received: 28 December 2025 Revised: 12 February 2026 Accepted: 26 February 2026 Published: 03 March 2026
MSC : 60B12, 62G30

This paper introduced a bivariate model constructed by coupling the iterated Farlie-Gumbel-Morgenstern (IFGM) copula with Chen marginals, yielding the bivariate IFGM-Chen distribution (IFGM-CD). The model was particularly tailored for reliability and survival analysis, as it accommodated bathtub-shaped hazard rates and captured a broader spectrum of dependence structures than traditional FGM-based models. Fundamental statistical properties were investigated, including product moments, conditional distributions, and reliability functions. A dedicated section on the stress-strength model within the IFGM-CD framework was provided, offering new insights into component reliability under dependent stress and strength. For parameter estimation, both maximum likelihood (ML) and Bayesian approaches were employed, supplemented by asymptotic and bootstrap confidence intervals. Extensive Monte Carlo simulations validated the performance of the estimators, and two real-data applications demonstrated the model's practicality and flexibility.
- iterated FGM copula,
- Chen distribution,
- bivariate lifetime model,
- stress-strength reliability,
- maximum likelihood estimation,
- Bayesian estimation,
- confidence intervals,
- bootstrap
Citation: M. A. Alawady, Mohamed A. Abd Elgawad, Salem A. Alyami, H. M. Barakat, I. A. Husseiny, G. M. Mansour, T. S. Taher, M. O. Mohamed. Bivariate Chen distribution based on iterated FGM copula: Properties and applications[J]. AIMS Mathematics, 2026, 11(3): 5379-5408. doi: 10.3934/math.2026222

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Abstract

This paper introduced a bivariate model constructed by coupling the iterated Farlie-Gumbel-Morgenstern (IFGM) copula with Chen marginals, yielding the bivariate IFGM-Chen distribution (IFGM-CD). The model was particularly tailored for reliability and survival analysis, as it accommodated bathtub-shaped hazard rates and captured a broader spectrum of dependence structures than traditional FGM-based models. Fundamental statistical properties were investigated, including product moments, conditional distributions, and reliability functions. A dedicated section on the stress-strength model within the IFGM-CD framework was provided, offering new insights into component reliability under dependent stress and strength. For parameter estimation, both maximum likelihood (ML) and Bayesian approaches were employed, supplemented by asymptotic and bootstrap confidence intervals. Extensive Monte Carlo simulations validated the performance of the estimators, and two real-data applications demonstrated the model's practicality and flexibility.

References

[1]	R. B. Nelsen, An Introduction to Copulas, New York: Springer-Verlag, 2006. https://doi.org/10.1007/0-387-28678-0
[2]	A. Sklar, Random variables, joint distributions functions, and copulas, Kybernetica, 9 (1973), 449–460.
[3]	A. K. Gupta, C. F. Wong, On three and five parameter bivariate beta distributions, Metrika, 32 (1985), 85–91. https://doi.org/10.1007/BF01897803 doi: 10.1007/BF01897803
[4]	M. K. H. Hassan, C. Chesneau, Bivariate generalized half-logistic distribution: Properties and its application in household financial affordability in KSA, Math. Comput. Appl., 27 (2022), 72. https://doi.org/10.3390/mca27040072 doi: 10.3390/mca27040072
[5]	S. O. Susam, A multi-parameter generalized Farlie-Gumbel-Morgenstern bivariate copula family via Bernstein polynomial, Hacettepe J. Math. Stat., 51 (2022), 618–631. https://doi.org/10.15672/hujms.993698 doi: 10.15672/hujms.993698
[6]	Y. Aliyu, U. Usman, On bivariate Nadarajah-Haghighi distribution derived from FGM copula in the presence of covariates, J. Nigerian Soc. Phys. Sci., 5 (2023), 871. https://doi.org/10.46481/jnsps.2023.871 doi: 10.46481/jnsps.2023.871
[7]	C. Blier-Wong, H. Cossette, E. Marceau, Exchangeable FGM copulas, Adv. Appl. Prob., 56 (2024), 205–234. https://doi.org/10.1017/apr.2023.19 doi: 10.1017/apr.2023.19
[8]	I. Dimitriou, On the overlap times in queues with dependence under a FGM copula, preprint paper, 2025. https://doi.org/10.48550/arXiv.2509.14100
[9]	G. M. Mansour, H. M. Barakat, I. A. Husseiny, M. Nagy, A. H. Mansi, M. A. Alawady, Measures of cumulative residual Tsallis entropy for concomitants of generalized order statistics based on the Morgenstern family with application to medical data, Math. Bio. Eng., 22 (2025a), 1572–1597. https://doi.org/10.3934/mbe.2025058 doi: 10.3934/mbe.2025058
[10]	J. S. Huang, S. Kotz, Correlation structure in iterated Farlie-Gumbel-Morgenstern distributions, Biometrika, 71 (1984), 633–636. https://doi.org/10.1093/biomet/71.3.633 doi: 10.1093/biomet/71.3.633
[11]	M. A. Abd Elgawad, H. M. Barakat, A. F. Hashem, G. M. Mansour, I. A. Husseiny, M. A. Alawady, et al., The bivariate Weibull distribution based on the GFGM copula, AIMS Math., 10 (2025), 27862–27897. https://doi.org/10.3934/math.20251224 doi: 10.3934/math.20251224
[12]	H. M. Barakat, E. M. Nigm, I. A. Husseiny, Measures of information in order statistics and their concomitants for the single iterated Farlie-Gumbel-Morgenstern bivariate distribution, Math. Popul. Stud., 28 (2021), 154–175. https://doi.org/10.1080/08898480.2020.1767926 doi: 10.1080/08898480.2020.1767926
[13]	H. M. Barakat, E. M. Nigm, M. A. Alawady, I. A. Husseiny, Concomitants of order statistics and record values from iterated FGM type bivariate-generalized exponential distribution, Revstat. Stat. J., 19 (2021), 291–307. https://doi.org/10.2991/jsta.d.190822.001 doi: 10.2991/jsta.d.190822.001
[14]	N. Chandra, A. James, F. Domma, H. Rehman, Bivariate iterated Farlie-Gumbel-Morgenstern stress-strength reliability model for Rayleigh margins: Properties and estimation, Stat. Theory Rel. Fields, 8 (2024), 315–334. https://doi.org/10.1080/24754269.2024.2398987 doi: 10.1080/24754269.2024.2398987
[15]	I. A. Husseiny, M. A. Alawady, S. A. Alyami, M. A. Abd Elgawad, Measures of extropy based on concomitants of generalized order statistics under a general framework from iterated Morgenstern family, Mathematics, 11 (2023), 1377. https://doi.org/10.3390/math11061377 doi: 10.3390/math11061377
[16]	D. Kundu, Bayesian inference and life testing plan for the Weibull distribution in presence of progressive censoring, Technometrics, 50 (2008), 144–154. https://doi.org/10.1198/004017008000000217 doi: 10.1198/004017008000000217
[17]	D. Kundu, B. Pradhan, Bayesian inference and life testing plans for generalized exponential distribution, Sci. China Ser. A: Math., 52 (2009), 1373–1388. https://doi.org/10.1007/s11425-009-0085-8 doi: 10.1007/s11425-009-0085-8
[18]	S. Singh, Y. M. Tripathi, S. J. Wu, On estimating parameters of a progressively censored lognormal distribution, J. Stat. Comput. Simul., 85 (2015), 1071–1089. https://doi.org/10.1080/00949655.2013.861838 doi: 10.1080/00949655.2013.861838
[19]	D. J. F. Davis, An analysis of some failure data, J. Amer. Stat. Ass., 47 (1952), 113–150. https://doi.org/10.1080/01621459.1952.10501160 doi: 10.1080/01621459.1952.10501160
[20]	H. W. Block, T. H. Savits, Burn-in, Statist. Sci., 12 (1997), 1–19. https://doi.org/10.1214/ss/1029963258 doi: 10.1214/ss/1029963258
[21]	M. Bebbington, C. D. Lai, R. Zitikis, Useful periods for lifetime distributions with bathtub shaped hazard rate functions, IEEE Trans. Reliab., 5 (2006), 245–251. https://doi.org/10.1109/TR.2001.874943 doi: 10.1109/TR.2001.874943
[22]	M. Bebbington, C. D. Lai, R. Zitikis, Modeling human mortality using mixtures of bathtub shaped failure distributions, J. Theor. Biol., 245 (2007), 528–538. https://doi.org/10.1016/j.jtbi.2006.11.011 doi: 10.1016/j.jtbi.2006.11.011
[23]	Z. Chen, A new two-parameter lifetime distribution with bathtub shape or increasing failure rate function, Stat. Prob. Lett., 49 (2000), 155–161. https://doi.org/10.1016/S0167-7152(00)00044-4 doi: 10.1016/S0167-7152(00)00044-4
[24]	E. S. A. El-Sherpieny, H. Z. Muhammed, E. M. Almetwally, Bivariate Chen distribution based on copula function: Properties and application of diabetic Nephropathy, J. Stat. Theory Pract., 16 (2022), 54. https://doi.org/10.1007/s42519-022-00275-7 doi: 10.1007/s42519-022-00275-7
[25]	N. Sreelakshmi, An introduction to copula-based bivariate reliability concepts, Commun. Stat. Theory Meth., 47 (2018), 996–1012. https://doi.org/10.1080/03610926.2017.1316396 doi: 10.1080/03610926.2017.1316396
[26]	M. A. Abd Elgawad, M. A. Alawady, H. M. Barakat, G. M. Mansour, I. A. Husseiny, A. F. Hashem, et al., Bivariate power Lomax Sarmanov distribution: Statistical properties, Reliability measures, and Parameter estimation, Alex. Eng. J., 113 (2025), 593–610. https://doi.org/10.1016/j.aej.2024.10.074 doi: 10.1016/j.aej.2024.10.074
[27]	S. Dey, S. Singh, Y. M. Tripathi, A. Asgharzadeh, Estimation and prediction for a progressively censored generalized inverted exponential distribution, Stat. Methodol., 32 (2016), 185–202. https://doi.org/10.1016/j.stamet.2016.05.007 doi: 10.1016/j.stamet.2016.05.007
[28]	G. M. Mansour, M. A. Abd Elgawad, A. S. Al-Moisheer, H. M. Barakat, M. A. Alawady, I. A. Husseiny, et al., Bivariate Epanechnikov-Weibull distribution based on sarmanov copula: Properties, simulation, and uncertainty measures with applications, AIMS Math., 10 (2025), 12689–12725. https://doi.org/10.3934/math.2025572 doi: 10.3934/math.2025572
[29]	X. Jia, D. Wang, P. Jiang, B. Guo, Inference on the reliability of weibull distribution with multiply type-I censored data, Reliab. Eng. Syst. Saf., 150 (2016), 171–181. https://doi.org/10.1016/j.ress.2016.01.025 doi: 10.1016/j.ress.2016.01.025
[30]	B. Efron, Bootstrap methods: Another look at the Jackknife, In: Kotz, S., Johnson, N. L. (eds) Breakthroughs in Statistics, New York: Springer, 1992. https://doi.org/10.1007/978-1-4612-4380-9-41
[31]	S. Kotz, Y. Lumelskii, M. Pensky, The Stress-Strength Model and Its Generalizations: Theory and Applications, New York: World Scientific, 2003. https://doi.org/10.1142/9789812564511-0001
[32]	M. A. Alawady, H. M. Barakat, T. S. Taher, I. A. Husseiny, Measures of extropy for $k$-record values and their concomitants based on Cambanis family, J. Stat. Theory Pract., 19 (2025), 11. https://doi.org/10.1007/s42519-024-00423-1 doi: 10.1007/s42519-024-00423-1
[33]	M. A. Alawady, H. M. Barakat, G. M. Mansour, I. A. Husseiny, Information measures and concomitants of $k-$record values based on Sarmanov family of bivariate distributions, Bull. Malays. Math. Sci. Soc., 46 (2023), 9. https://doi.org/10.1007/s40840-022-01396-9 doi: 10.1007/s40840-022-01396-9
[34]	H. M. Barakat, M. A. Alawady, T. S. Taher, I. A. Husseiny, Second-order concomitants of order statistics from bivariate Cambanis family: Some information measures–estimation, Commun. Stat. Simul. Comput., 54 (2025), 3195–3221. https://doi.org/10.1080/03610918.2024.2344023 doi: 10.1080/03610918.2024.2344023
[35]	I. A. Husseiny, H. M. Barakat, T. S. Taher, M. A. Alawady, Fisher information in order statistics and their concomitants for Cambanis bivariate distribution, Math. Slovaca, 74 (2024), 501–520. https://doi.org/10.1515/ms-2024-0038 doi: 10.1515/ms-2024-0038
[36]	G. M. Mansour, H. M. Barakat, M. A. Alawady, M. A. Abd Elgawad, H. Alqifari, T. S. Taher, et al., Uncertainty measures for concomitants of upper k-record values based on the Huang-Kotz-Morgenstern type Ⅱ family, AIMS Math., 10 (2025), 29071–29106. https://doi.org/10.3934/math.20251279 doi: 10.3934/math.20251279
[37]	V. Barnett, Some bivariate uniform distributions, Commun. Stat. Theory Meth., 9 (1980), 453–461. https://doi.org/10.1080/03610928008827893 doi: 10.1080/03610928008827893
[38]	D. J. Hand, A Handbook of Small Data Sets, Boca Raton: Chapman & Hall/CRC, 1993.
[39]	T. Saali, M. Mesfioui, A. Shabri, Multivariate extension of Raftery copula, Mathematics, 11 (2023), 414. https://doi.org/10.3390/math11020414 doi: 10.3390/math11020414
[40]	S. G. Meintanis, Test of fit for Marshall-Olkin distributions with applications, J. Stat. Plan. Inf., 137 (2007), 3954–3963. https://doi.org/10.1016/j.jspi.2007.04.013 doi: 10.1016/j.jspi.2007.04.013
[41]	S. K. Khames, N. A. Mokhlis, Bivariate general exponential models with stress-strength reliability application, J. Egyp. Math. Soc., 28 (2020), 9. https://doi.org/10.1186/s42787-020-0069-y doi: 10.1186/s42787-020-0069-y
[42]	M. El-Morshedy, M. S. Eliwa, M. H. Tahir, M. Alizadeh, R. El-Desokey, A. Al-Bossly, et al., A bivariate extension to exponentiated inverse flexible Weibull distribution: Shock model, features, and inference to model asymmetric data, Symmetry, 15 (2023), 411. https://doi.org/10.3390/sym15020411 doi: 10.3390/sym15020411

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