Journal of the Iranian Statistical Society

Stress-Strength and Ageing Intensity Analysis via a New Bivariate Negative Gompertz-Makeham Model

Authors

Department of Statistics, Ordered Data, Reliability and Dependency Center of Excellence, Ferdowsi University of Mashhad, Mashhad, Iran

10.52547/jirss.20.1.219

Abstract

In Demography and modelling mortality (or failure) data the univariate Makeham-Gompertz is well-known for its extension of exponential distribution. Here, a bivariate class of Gompertz--Makeham distribution is constructed based on random number of extremal events. Some reliability properties such as ageing intensity, stress-strength based on competing risks are given. Also dependence properties such as dependence structure, association measures and tail dependence measures are obtained. A simulation study and a performance analysis is given based on estimators such as MLE, Tau-inversion and Rho-inversion.

Keywords

  1. Abd El-Bar, A. M. (2018), An extended gompertz-makeham distribution with application to lifetime data. Communications in Statistics-Simulation and Computation, 47(8), 2454-2475. [DOI:10.1080/03610918.2017.1348517]
  2. Adham, S. A., and Walker, S. G. (2001), A multivariate Gompertz-type distribution. Journal of Applied Statistics. 28(8), 1051-1065. [DOI:10.1080/02664760120076706]
  3. Anderson, J. E., Louis, T. A., Holm, N. V., and Harvald, B. (1992), Time-dependent association measures for bivariate survival distributions. Journal of the American Statistical Association, 87(419), 641-650. [DOI:10.1080/01621459.1992.10475263]
  4. Bailey, R. C., and Homer, L. D. (1977), Computations for a best match strategy for kidney transplantation. Transplantation, 23(4), 329-336. [DOI:10.1097/00007890-197704000-00006]
  5. Bailey, R. C., Homer, L. D., and Summe, J. P. (1977), A proposal for the analysis of kidney graft survival. Transplantation, 24(5), 309-315. [DOI:10.1097/00007890-197711000-00001]
  6. Bebbington, M., Green, R., Lai, C. D., and Zitikis, R. (2014), Beyond the Gompertz law: exploring the late-life mortality deceleration phenomenon. Scandinavian Actuarial Journal, 2014(3), 189-207. [DOI:10.1080/03461238.2012.676562]
  7. Clayton, D. G. (1978), A model for association in bivariate life tables and its application in epidemiological studies of familial tendency in chronic disease incidence. Biometrika, 65(1), 141-151. [DOI:10.1093/biomet/65.1.141]
  8. Denuit, M., Dhaene, J., Goovaerts, M., and Kaas, R. (2006), Actuarial theory for dependent risks: measures, orders and models. John Wiley & Sons.
  9. El-Sherpieny, E. A., Ibrahim, S. A., and Bedar, R. E. (2013), A new bivariate distribution with generalized Gompertz marginals. Asian Journal of Applied Sciences, 1(04), 141-150.
  10. Feng, X., and He, G. (2008), Estimation of parameters of the Makeham distribution using the least squares method. Mathematics and Computers in Simulation , 77(1), 34-44. [DOI:10.1016/j.matcom.2007.01.009]
  11. Golubev, A. (2009), How could the Gompertz--Makeham law evolve. Journal of theoretical Biology, 258(1), 1-17. [DOI:10.1016/j.jtbi.2009.01.009]
  12. Gompertz, B. (1825), XXIV. On the nature of the function expressive of the law of human mortality, and on a new mode of determining the value of life contingencies. In a letter to Francis Baily, Esq. FRS & c. Philosophical transactions of the Royal Society of London, 115, 513-583. [DOI:10.1098/rstl.1825.0026]
  13. Jiang, R., Ji, P., and Xiao, X. (2003), Aging property of unimodal failure rate models. Reliability Engineering & System Safety, 79(1), 113-116. [DOI:10.1016/S0951-8320(02)00175-8]
  14. Johnson, N. L., and Kotz, S. (1975), A vector multivariate hazard rate. Journal of Multivariate Analysis, 5(1), 53-66. [DOI:10.1016/0047-259X(75)90055-X]
  15. Johnson, N., Kotz, S., and Balakrishnan, N. (1994), Continuous Univariate Distributions. Volume 1, 2nd Edition. John Wiley and Sons, New York.
  16. Johnson, N. L., Kotz, S., and Balakrishnan, N. (1997), Discrete multivariate distributions. Volume 165. Wiley. John Wiley and Sons (New York).
  17. Juckett, D. A., and Rosenberg, B. (1993), Comparison of the Gompertz and Weibull functions as descriptors for human mortality distributions and their intersections. Mechanisms of Ageing and Development, 69(1-2), 1-31. [DOI:10.1016/0047-6374(93)90068-3]
  18. Kolev, N. (2016), Characterizations of the class of bivariate Gompertz distributions. Journal of Multivariate Analysis, 148, 173-179. [DOI:10.1016/j.jmva.2016.03.004]
  19. Lai, C. D., and Xie, M. (2006), Stochastic ageing and dependence for reliability . Springer Science & Business Media.
  20. Marshall, A. W., and Olkin, I. (1997), A new method for adding a parameter to a family of distributions with application to the exponential and Weibull families. Biometrika, 84(3), 641-652. [DOI:10.1093/biomet/84.3.641]
  21. Marshall, A. W., and Olkin, I. (2007), Life distributions. Springer, New York.
  22. Marshall, A. W., and Olkin, I. (2015), A bivariate Gompertz--Makeham life distribution. Journal of Multivariate Analysis, 139, 219-226. [DOI:10.1016/j.jmva.2015.02.011]
  23. Melnikov, A., and Romaniuk, Y. (2006), Evaluating the performance of Gompertz, Makeham and Lee-Carter mortality models for risk management with unit-linked contracts. Insurance: Mathematics and Economics, 39(3), 310-329. [DOI:10.1016/j.insmatheco.2006.02.012]
  24. Missov, T. I., and Lenart, A. (2013), Gompertz--Makeham life expectancies: expressions and applications. Theoretical Population Biology, 90, 29-35. [DOI:10.1016/j.tpb.2013.09.013]
  25. Nelsen, R. B. (2007), An introduction to copulas. Springer Science & Business Media.
  26. Oakes, D. (1989), Bivariate survival models induced by frailties. Journal of the American Statistical Association, 84(406), 487-493. [DOI:10.1080/01621459.1989.10478795]
  27. Scollnik, D. P. M. (1995), Simulating random variates from Makeham's distribution and from others with exact or nearly log--concave densities. Transactions of the Society of Actuaries, 47, 409-437.
  28. Shih, J. H., and Emura, T. (2019), Bivariate dependence measures and bivariate competing risks models under the generalized FGM copula. Statistical Papers, 60(4), 1101-1118. [DOI:10.1007/s00362-016-0865-5]
  29. Szymkowiak, M. (2020), Lifetime Analysis by Aging Intensity Functions. Springer International Publishing.
  30. Wang, J. L., Muller, H. G., and Capra, W. B. (1998), Analysis of oldest-old mortality: Life--tables revisited. The Annals of Statistics, 26(1), 126-163.
Journal of the Iranian Statistical Society
Volume 20, Issue 1
June 2021
Pages 219-246
  • Receive Date: 23 July 2022
  • Revise Date: 19 May 2024
  • Accept Date: 23 July 2022