M. Esfahani - Department of Statistics, Faculty of Mathematical Sciences, Ferdowsi University of Mashhad, Iran
M. Amini - Department of Statistics, Faculty of Mathematical Sciences, Ferdowsi University of Mashhad, Iran
G. R. Mohtashami-Borzadaran - Department of Statistics, Faculty of Mathematical Sciences, Ferdowsi University of Mashhad, Iran
A. Dolati - Department of Statistics, Faculty of Mathematical Sciences, Ferdowsi University of Mashhad, Iran

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-۱۹, the appropriateness of the proposed model is examined.

Copula function, bivariate Gompertz--Makeham distribution, dependence measures, dependence structure, reliability