In this paper, we propose bivariate iterated Farlie–Gumbel–Morgenstern (FGM) due to [Huang and Kotz (1984). Correlation structure in iterated Farlie-Gumbel-Morgenstern distributions. Biometrika 71(3), 633–636. https://doi.org/10.2307/2336577] with Rayleigh marginals. The dependence stress–strength reliability function is derived with its important reliability characteristics. Estimates of dependence reliability parameters are obtained. We analyse the effects of dependence parameters on the reliability function. We found that the upper bound of the positive correlation coefficient is attaining to 0.41 under a single iteration with Rayleigh marginals. A comprehensive comparison between classical FGM with iterated FGM copulas is graphically examined to assess the over or under estimation of reliability with respect to α and β. We propose a two-phase estimation procedure for estimating the reliability parameters. A Monte-Carlo simulation study is conducted to assess the finite sample behaviour of the proposed reliability estimators. Finally, the proposed estimators are examined and validated with real data sets.

References

• Ahmed, D., Khames, S., & Nahed, M. (2020). Inference for stress–strength models based on the bivariate general Farlie–Gumbel–Morgenstern distributions. Journal of Statistics Applications & Probability Letters, 7(open in a new window)(3(open in a new window)), 141–150. http://dx.doi.org/10.18576/jsapl/070304 
• Almetwally, E. M., Muhammed, H. Z., & El-Sherpieny, E. -S. A. (2020). Bivariate Weibull distribution: Properties and different methods of estimation. Annals of Data Science, 7(open in a new window)(1(open in a new window)), 163–193. https://doi.org/10.1007/s40745-019-00197-5 
• Bai, X., Li, X., Balakrishnan, N., & He, M. (2021). Statistical inference for dependent stress–strength reliability of multi-state system using generalized survival signature. Journal of Computational and Applied Mathematics, 390(open in a new window), 113316. https://doi.org/10.1016/j.cam.2020.113316 
• Bairamov, I., & Kotz, S. (2002). Dependence structure and symmetry of Huang–Kotz FGM distributions and their extensions. Metrika, 56(open in a new window)(1(open in a new window)), 55–72. https://doi.org/10.1007/s001840100158 
• Barbiero, A. (2017). Assessing how the dependence structure affects the reliability parameter of the strength–stress model. Eccomas Proceedia UNCECOMP, 640–650. https://doi.org/10.7712/120217.5399.16870. 
• Basu, A. (1971). Bivariate failure rate. Journal of the American Statistical Association, 66(open in a new window)(333(open in a new window)), 103–104. https://doi.org/10.1080/01621459.1971.10482228 
• Bekrizadeh, H., Parham, G. A., & Jamshidi, B. (2017). A new asymmetric class of bivariate copulas for modeling dependence. Communications in Statistics-Simulation and Computation, 46(open in a new window)(7(open in a new window)), 5594–5609. https://doi.org/10.1080/03610918.2016.1169292 
• Bekrizadeh, H., Parham, G. A., & Zadkarmi, M. R. (2012). The new generalization of Farlie–Gumbel–Morgenstern copulas. Applied Mathematical Sciences, 6(71), 3527–3533. 
• Birnbaum, Z. W. (1956). On a use of mann-whitney statistics. Proceeding Third Berkley Symposium on Mathematical Statistics and Probability, 1, 13–17. 
• Chandra, N., & Pandey, M. (2012). Bayesian reliability estimation of bivariate Marshal–Olkin exponential stress–strength model. International Journal of Reliability and Applications, 13(1), 37–47. 
• Chandra, N., & Rathaur, V. K. (2017). Bayes estimation of augmenting gamma strength reliability of a system under non-informative prior distributions. Calcutta Statistical Association Bulletin, 69(open in a new window)(1(open in a new window)), 87–102. https://doi.org/10.1177/0008068317696574 
• Chandra, N., & Rathaur, V. K. (2020). Inferences on non-identical stress and generalized augmented strength reliability parameters under informative priors. International Journal of Reliability, Quality and Safety Engineering, 27(open in a new window)(04(open in a new window)), 2050014. https://doi.org/10.1142/S021853932050014X 
• Domma, F., & Giordano, S. (2012). A stress–strength model with dependent variables to measure household financial fragility. Statistical Methods & Applications, 21(open in a new window)(3(open in a new window)), 375–389. https://doi.org/10.1007/s10260-012-0192-5 
• Domma, F., & Giordano, S. (2013). A copula-based approach to account for dependence in stress–strength models. Statistical Papers, 54(open in a new window)(3(open in a new window)), 807–826. https://doi.org/10.1007/s00362-012-0463-0 
• Durante, F. (2006). A new class of symmetric bivariate copulas. Journal of Nonparametric Statistics, 18(open in a new window)(7–8(open in a new window)), 499–510. https://doi.org/10.1080/10485250701262242 
• Farlie, D. J. (1960). The performance of some correlation coefficients for a general bivariate distribution. Biometrika, 47(open in a new window)(3/4(open in a new window)), 307–323. https://doi.org/10.2307/2333302 
• Gumbel, E. J. (1960). Bivariate exponential distributions. Journal of the American Statistical Association, 55(open in a new window)(292(open in a new window)), 698–707. https://doi.org/10.1080/01621459.1960.10483368 
• Gupta, P. L., & Gupta, R. C. (2012). Some properties of the bivariate lognormal distribution for reliability applications. Applied Stochastic Models in Business and Industry, 28(open in a new window)(6(open in a new window)), 598–606. https://doi.org/10.1002/asmb.935 
• Huang, J., & Kotz, S. (1984). Correlation structure in iterated Farlie–Gumbel–Morgenstern distributions. Biometrika, 71(open in a new window)(3(open in a new window)), 633–636. https://doi.org/10.2307/2336577 
• Huang, J. S., & Kotz, S. (1999). Modifications of the Farlie–Gumbel–Morgenstern distributions. A tough hill to climb. Metrika, 49(open in a new window)(2(open in a new window)), 135–145. https://doi.org/10.1007/s001840050030 
• James, A., & Chandra, N. (2022). Dependence stress–strength reliability estimation of bivariate xgamma exponential distribution under copula approach. Palestine Journal of Mathematics, 11, 213–233. 
• James, A., Chandra, N., & Pandey, M. (2022). A copula based stress–strength reliability estimation with Lindley marginals. Journal of Reliability and Statistical Studies, 15(open in a new window)(1(open in a new window)), 341–380. https://doi.org/10.13052/jrss0974-8024.15114 
• James, A., Chandra, N., & Sebastian, N. (2023). Stress–strength reliability estimation for bivariate copula function with Rayleigh marginals. International Journal of System Assurance Engineering and Management, 14(open in a new window)(1(open in a new window)), 1–20. https://doi.org/10.1007/s13198-022-01836-6 
• Johnson, N. L., & Kotz, S. (1975a). On some generalized Farlie–Gumbel–Morgenstern distributions. Communications in Statistics-Theory and Methods, 4(open in a new window)(5(open in a new window)), 415–427. https://doi.org/10.1080/03610927708827509 
• Johnson, N. L., & Kotz, S. (1975b). A vector multivariate hazard rate. Journal of Multivariate Analysis, 5(open in a new window)(1(open in a new window)), 53–66. https://doi.org/10.1016/0047-259X(75)90055-X 
• Johnson, N. L., & Kotz, S. (1977). On some generalized Farlie–Gumbel–Morgenstern distributions-II regression, correlation and further generalizations. Communications in Statistics-Theory and Methods, 6(open in a new window)(6(open in a new window)), 485–496. https://doi.org/10.1080/03610927708827509 
• Kotz, S., & Pensky, M. (2003). The Stress–Strength Model and its Generalizations: Theory and Applications. World Scientific. https://doi.org/10.1142/9789812564511_0001 
• Lutfiah, I. A., Elaal, K. M., & Jarwan, R. S. (2017). Inference of bivariate generalized exponential distribution based on copula functions. Applied Mathematical Sciences, 11(open in a new window)(24(open in a new window)), 1155–1186. https://doi.org/10.12988/ams.2017.7398 
• Mokhlis, N. A., Ibrahim, E. J., & Gharieb, D. M. (2017). Stress strength reliability with general form distributions. Communications in Statistics Theory and Methods, 46(open in a new window)(3(open in a new window)), 1230–1246. https://doi.org/10.1080/03610926.2015.1014110 
• Morgenstern, D. (1956). Einfache Beispiele zweidimensionaler Verteilungen. Mitteilingsblatt fur Mathematische Statistik, 8, 234–235. 
• Nadarajah, S. (2005). Reliability for some bivariate gamma distributions. Mathematical Problems in Engineering, 2005(open in a new window)(2(open in a new window)), 151–163. https://doi.org/10.1155/MPE.2005.151 
• Pak, A., Raqab, M. Z., Mahmoudi, M. R., Band, S. S., & Mosavi, A. (2022). Estimation of stress–strength reliability R=P(X>Y) based on Weibull record data in the presence of inter-record times. Alexandria Engineering Journal, 61(open in a new window)(3(open in a new window)), 2130–2144. https://doi.org/10.1016/j.aej.2021.07.025 
• Patil, D. D., & Naik-Nimbalkar, U. V. (2017). Computation and estimation of reliability for some bivariate copulas with Pareto marginals. Journal of Statistical Computation and Simulation, 87(open in a new window)(18(open in a new window)), 3563–3589. https://doi.org/10.1080/00949655.2017.1376328 
• Pham, H. (2020). On stress–strength interval-system reliability with applications in heart conditions. International Journal of Mathematical, Engineering and Management Sciences, 5(open in a new window)(1(open in a new window)), 1–12. https://doi.org/10.33889/IJMEMS.2020.5.1.001 
• Rubio, F. J., & Steel, M. F. (2013). Bayesian inference for P(X<Y) using asymmetric dependent distributions. Bayesian Analysis, 8(open in a new window)(1(open in a new window)), 43–62. https://doi.org/10.1214/13-BA802 
• Sankaran, P. G., & Nair, N. U. (1991). On bivariate vitality functions. Proceeding of National Symposium on Distribution Theory, 11, 189–211. 
• Shanbhag, D. N., & Kotz, S. (1987). Some new approaches to multivariate probability distributions. Journal of Multivariate Analysis, 22(open in a new window)(2(open in a new window)), 189–211. https://doi.org/10.1016/0047-259X(87)90085-6 

To cite this article: N. Chandra, A. James, Filippo Domma & Habbiburr Rehman (03 Oct 2024): Bivariate iterated Farlie–Gumbel–Morgenstern stress–strength reliability model for Rayleigh margins: Properties and estimation, Statistical Theory and Related Fields, DOI: 10.1080/24754269.2024.2398987

To link to this article: https://doi.org/10.1080/24754269.2024.2398987