Statistical inference for the non-conforming rate of FGM Copula-Based bivariate exponential lifetime

Document Type : Research Paper


1 Khorasan Razavi Agricultural and Natural Resources Research and Education Center, AREEO, Mashhad, Iran

2 Department of Statistics, Faculty of Mathematical Sciences, Ferdowsi University of Mashhad, Mashhad, Iran

3 Department of Mathematics, Faculty of Mathematical Sciences, University of Mazandaran, Babolsar, Iran


‎‎Lifetime performance index ‎is widely used as process capability index to evaluate the performance and potential of a process‎. ‎In manufacturing industries‎, ‎the lifetime of a product is considered to be conforming if it exceeds a given lower threshold value‎, ‎so‎ nonconforming products are those that fail to exceed this value.‏ Nonconformities are ‎so ‏‎important‎ that affect the safe or effective use of the products. ‏‎This article deals with ‎the processes‎ that ‎the ‎products' ‎lifetime is related to a two-component system, ‎distributed ‎as Farlie-Gumbel-Morgenstern (FGM) copula-based bivariate ‎exponential‎ ‎and ‎presen‏‎ts‎‎ the ‎probability ‎of ‎non-conforming ‎products‎. Also, bootstrap upper confidence bounds are constructed and their performance are investigated in simulation study. In addition, Monte Carlo scheme is applied to do hypothesis testing on it. Finally, two example sets are presented to demonstrate the application of the proposed index.‎


[1] M. V. Ahmadi, M. Doostparast, J. Ahmadi, Statistical inference for the lifetime performance index based on generalised order statistics from exponential distribution, International Journal of Systems Science. 46(6) (2015) 1094-1107.
[2] M. V. Ahmadi, J. Ahmadi. M. Abdi, Evaluating the lifetime performance index of products based on generalized order  statistics from two-parameter exponential model, International Journal of System Assurance Engineering and  Management. 10 (2019) 251-275.
[3] H. T. Chen, L. I. Tong, K. S. Chen, Assessing the lifetime performance of electronic components by con dence interval, Journal of the Chinese Institute of Industrial Engineers. 19 (2002) 5360.
[4] N. Clark, M. Dabkowski, P. J. Driscoll, D. Kennedy, I. Kloo, H. Shi, Empirical decision rules for improving the uncertainty reporting of small sample system usability scale scores, International Journal of Human-Computer Interaction, 37 (13) (2021) 1191-1206.
[5] F. M. Dekking, C. Kraaikamp, H. P. Lopuhaa, L. E. Meester, A Modern Introduction to Probabitity and Statistics. Springer, Verlag, London, 2005.
[6] B. Efron, Bootstrap methods: another look at the jacknife. The Annals of Statistics. 7(1) (1979) 1-26.
[7] B. Efron, Nonparametric standard errors and con dence intervals, The Canadian Journal of Statistics. 9(2) (1981)  139-172.
[8] B. Efron, The jacknife, the bootstrap and other resampling plans. In Regional Conference Series in Applied  Mathematics, Philadelphia: SIAM, 1982.
[9] B. Efron, Bootstrap con dence intervals for a class of parametric problems, Biometrika. 72 (1985) 45-58.
[10] B. Efron, R. Tibshirani, Bootstrap methods for standard errors, con dence intervals, and other measures of statistical  accuracy, Statistical Science. 1(1) (1986) 54-75.
[11] B. Efron, Better bootstrap con dence intervals, Journal of the American Statistical Association. 82 (1987) 171-185.
[12] N. I. Fisher, P. Switzer, Chi-plots for assessing dependence, Biometrika. 72(2) (1985) 253-265.
[13] N. I. Fisher, P. Switzer, Graphical assessment of dependence: Is a picture worth 100 tests?, The American Statistician. 55(3) (2001) 233-239.
[14] C. Genest, J. MacKay, The Joy of Copulas: Bivariate Distributions with Uniform Marginals (Com: 87V41 P248), The American Statistician. 40 (1986) 280-283.
[15] C. Genest, L. P. Rivest, Statistical inference procedures for bivariate Archimedean copulas, Journal of the American Statistical Association. 88 (1993) 1034-1043.
[16] C. Genest, J. C. Boies, Detecting dependence with Kendall plots, The American Statistician. 57(4) (2003) 275-284.
[17] C. Genest, J. F. Quessy, B. Remillard, Goodness-of- t procedures for copula models based on the probability integral transformation, Scandinavian Journal of Statistics. 33(2) (2006) 337-366.
[18] C. Genest, B. Remillard, Validity of the parametric bootstrap for goodness-of- t testing in semiparametric models, Annales de 1'Institut Henri Poincare- Probabilites et Statistiques. 44(6) (2008) 1096-1127.
[19] C. Genest, A. C. Favre, Everything You Always Wanted to Know about Copula Modeling but Were Afraid to Ask, Journal of Hydrologic Engineering. 12 (2007) 347-368.
[20] S. S. Hashemi-Bosra, E. Salehi, The mean residual lifetime of parallel systems with two exchangeable components under the generalized Farlie-Gumbel-Morgenstern model, Journal of Mahani Mathematical Research Center, 2 (2) (2013) 61-72.
[21] H. Joe, Multivariate Models and Dependence Concepts, Chapman & Hall Ltd, 1997.
[22] M. E. Johnson, Multivariate Statistical Simulation. Wiley, New York, 1987.
[23] M. Khademi, V. Amirzadeh, Process control using assumed fuzzy test and fuzzy acceptance region, Journal of Mahani Mathematical Research Center, 2 (2) (2013) 29-37.
[24] J. E. Lawless, Statistical Models and Methods for Lifetime Data, John Wiley and Sons Inc., New York, 1982.
[25] W. C. Lee, J. W. Wu, C. W. Hong, Assessing the lifetime performance index of products with the exponential  distribution under progressively type II right censored samples, Journal of Computational and Applied Mathematics. 231 (2009) 648-656.
[26] E. L. Lehmann, G. Casella, Theory of Point Estimation. 2nd ed. New York: Springer, 1998.
[27] S. Maiti, A. Bhattacharya, M. Saha, On generalizing lifetime performance index, Life Cycle Reliability and Safety Engineering. 10 (2021) 31-38.
[28] D.C. Montgomery, Introduction To Statistical Quality Control, John Wiley & Sons, New York, NY, USA, 1985.
[29] R.B. Nelsen, An Introduction to Copulas, Springer-Verlag, 1999.
[30] J. Orlo , J. Bloom, Bootstrap con dence intervals, Class 24, 18.05, Springer, Massachusetts Institute of Technology: MIT OpenCourseWare, 2017,
[31] W. L. Pearn, M-H Shu, Manufacturing capability control for multiple power-distribution switch processes based on modi ed Cpk MPPAC , Microelectronics Reliability. 43 (2003) 963-975.
[32] S. M. Ross, Introduction to probability and statistics for engineers and scientists (4th ed.). Associated Press, 2009.
[33] A.W. Sklar, Fonctions de repartition a n dimension et leurs marges, Publications de l'Institut de Statistique de  l'Universite de Paris. 8 (1959) 229-231.
[34] A. A-E.Soliman, E. A-S. Ahmed, A. H. Abd Ellah, A-W. A. Farghal, Assessing the lifetime performance index using exponentiated Frechet distribution with the progressive  rst-failure-censoring scheme, American Journal of Theoretical  and Applied Statistics. 3(6) (2014) 167-176.
[35] L. I. Tong, K. S. Chen, H. T. Chen, Statistical testing for assessing the performance of lifetime index of electronic  components with exponential distribution, Journal of Quality Reliability Management. 19 (2002) 812-824.
[36] W. Wang, M. T. Wells, Model selection and semiparametric inference for bivariate failure-time data (with discussion),  Journal of the American Statistical Association. 95(1) (2000) 62-76.
[37] S. Wang, X. Zhang, L. Liu, Multiple stochastic correlations modeling for microgrid reliability and economic evaluation using pair-copula function, International Journal of Electrical Power & Energy Systems. 76 (2016) 44-52.
[38] A. Wiboonpongse, J. Liu, S. Sriboonchitta, T. Denoeux, Modeling dependence between error components of the stochastic frontier model using copula: Application to intercrop co ee production in Northern Thailand, International Journal of Approximate Reasoning. 65 (2015) 34-44.
[39] J. W. Wu, C. W. Hong, W. C. Lee, Computational procedure of lifetime performance index of products for the Burr XII distribution with upper record values, Applied Mathematics and Computation. 227 (2014) 701-716.
[40] J. Yan, Multivariate Modeling with Copulas and Engineering Applications. In H Pham (ed.), Handbook in Engineering Statistics, Springer-Verlag, 2006, pp. 973990.
[41] L. Yang, X. J. Cai, M. Li, S. Hamori, Modeling dependence structures among international stock markets: Evidence from hierarchical Archimedean copulas, Economic Modelling. 51 (2015) 308-314.
[42] K. S. Zhang, J. G. Lin, P. R. Xu, A new class of copulas involving geometric distribution: Estimation and applications, Insurance: Mathematics and Economics. 66 (2016) 1-10.
  • Receive Date: 22 July 2021
  • Revise Date: 11 September 2021
  • Accept Date: 26 September 2021
  • First Publish Date: 19 October 2021