A new copula-based bivariate Gompertz--Makeham model and its application to COVID-19 mortality data

Document Type : Research Paper

Authors

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

2 Department of Statistics, Ordered ِData, Reliability and Dependency Center of Excellence, Ferdowsi University of Mashhad, P.O. Box 91775-1159, Mashhad, Iran

3 Department of Statistics, Faculty of Mathematical Sciences, Yazd University, Yazd, Iran

Abstract

One of the useful distributions in modeling mortality (or failure) data is the univariate Gompertz--Makeham distribution. To examine the relationship between the two variables, the extended bivariate Gompertz--Makeham distribution is introduced, and its properties are provided. Also, some reliability indices, including aging intensity and stress-strength reliability, are calculated for the proposed model. Here, a new copula function is constructed based on the extended bivariate Gompertz--Makeham  distribution. Some of its features including dependency properties, such as dependence structure, some  measures of dependence, and tail dependence,  are studied.
The estimation of the  parameters of new copula is presented, and at the end, a simulation study and a performance analysis based on the real data are presented.  So, by analyzing the mortality data due to COVID-19, the appropriateness of the proposed model is examined.

Keywords

References

[1] A. Algarni, A. Almarashi, I. Elbatal, A. Hassan, E. M. Almetwally, A. Daghistani, M. Elgarhy, Type I half logistic
Burr XG family: Properties, Bayesian, and non-Bayesian estimation under censored samples and applications to
COVID-19 data, Mathematical Problems in Engineering, 2021.
[2] A. Dolati, J. F. Sanchez, M. Ubeda-Flores, A copula-based family of fuzzy implication operators, Fuzzy Sets and
Systems, 211 (2013), 55-61.
[3] N. Doodman, M. Amini, H. Jabbari, A. Dolati, FGM generated Archimedean copulas with concave multiplicative
generators, Iranian Journal of Fuzzy Systems, 18(2) (2021), 15-29.
[4] C. Genest, B. Remillard, D. Beaudoin, Goodness-of-fit tests for copulas: A review and a power study, Insurance:
Mathematics and Economics, 42 (2008), 199-213.
[5] P. Grzegorzewski, Probabilistic implications, in: EUSFLAT-2011 and LFA-2011, Amsterdam, (2011), 254-258.
[6] P. H´ajek, R. Mesiar, On copulas, quasicopulas and fuzzy logic, Soft Computing, 12(12) (2008), 1239-1243.
[7] P. Helbin, M. Baczynski, P. Grzegorzewski, W. Niemyska, Some properties of fuzzy implications based on copulas, Information Sciences, 502 (2019), 1-17.
[8] R. Jiang, P. Ji, X. Xiao, Aging property of unimodal failure rate models, Reliability Engineering and System Safety, 79(1) (2003), 113-116.
[9] H. Joe, Dependence modeling with copulas, CRC Press, 2014.
[10] H. Joe, J. J. Xu, The estimation method of inference functions for margins for multivariate models, Technical
Report: Department of Statistics University of British Columbia, (1996), 166.
[11] M. E. Johnson, Multivariate statistical simulation distributions, New York, John Wiley and Sons, Statistics and
Probability Letters, 46(4) (1987), 359-364.
[12] N. Kolev, Characterizations of the class of bivariate Gompertz distributions, Journal of Multivariate Analysis, 148
(2016), 173-179.
[13] S. Kotz, M. Pensky, The stress-strength model and its generalizations: Theory and applications, World Scientific, 2003.
[14] C. D. Lai, M. Xie, Stochastic ageing and dependence for reliability, Springer Science and Business Media, New
York, 2006.
[15] W. M. Makeham, On the law of mortality and the construction of annuity tables, Journal of the Institute of
Actuaries and Assurance Magazine, 8 (1860), 301-310.
[16] A. W. Marshall, I. Olkin, A bivariate Gompertz–Makeham life distribution, Journal of Multivariate Analysis, 139
(2015), 219-226.
[17] H. A. Mohtashami Borzadaran, H. Jabbari, M. Amini, A. Dolati, Stress-strength and ageing intensity analysis via a
new bivariate negative Gompertz-Makeham model, Journal of The Iranian Statistical Society, 20(1) (2021), 219-246.
[18] R. B. Nelsen, An introduction to copulas, Springer-Verlag, New York, 2006.
[19] B. P. Rao, On some analogues of lack of memory properties for the Gompertz distribution, Communications in
Statistics-Theory and Methods, 47(18) (2018), 4415-4421.
[20] J. H. Shih, T. Emura, Bivariate dependence measures and bivariate competing risks models under the generalized
FGM copula, Statistical Papers, 60(4) (2019), 1101-1118.
[21] M. Sklar, Fonctions de repartition an dimensions et leurs marges, Publications de l’Institut de Statistique de
l’Universit de Paris, 8 (1959), 229-231.
[22] C. Sun, Z. Bie, M. Xie, J. Jiang, Fuzzy copula model for wind speed correlation and its application in wind
curtailment evaluation, Renewable Energy, 93 (2016), 68-76.
[23] W. S. B. Woolhouse, Observations on Gompertzs law of mortality and dependence between it and Simpson’s rule for
finding the value of an annuity on three lives, The Assurance Magazine, and Journal of the Institute of Actuaries, 10 (1863), 121-130.
[24] R. R. Yager, Modeling holistic fuzzy implication using copulas, Fuzzy Optimization and Decision Making, 5(3) (2006), 207-226.
Iranian Journal of Fuzzy Systems
Volume 20, Issue 3
May and June 2023
Pages 159-175

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