Abstract
Abstract The use of the exponential distribution and its multivariate generalizations is extremely popular in lifetime modeling. Freund’s bivariate exponential model (1961) is based on the idea that the remaining lifetime of any entity in a bivariate system is shortened when the other entity defaults. Such a model can be quite useful for studying systemic risk, for instance in financial systems. Guzmics and Pflug (2019) revisited Freund’s model, deriving the corresponding bivariate copula and examined some characteristics of it; furthermore, we opened the door for a multivariate setting. Now we present further investigations in the bivariate model: we compute the tail dependence coefficients, we examine the marginal and joint distributions of the componentwise maxima, which leads to an extreme value copula, which – to the best of our knowledge – has not been investigated in the literature yet. The original bivariate model of Freund has been extended to more variables by several authors. We also turn to the multivariate setting, and our focus is different from that of the previous generalizations, and therefore it is novel: examining the distribution of the sum and of the average of the lifetime variables (provided that the shock parameters are all the same) leads to new families of univariate distributions, which we call Exponential Gamma Mixture Type I and Type II (EGM) distributions. We present their basic properties, we provide asymptotics for them, and finally we also provide the limiting distribution for the EGM Type II distribution.
Highlights
We consider the bivariate lifetime model introduced by Freund [4]
Guzmics and P ug (2019) revisited Freund’s model, deriving the corresponding bivariate copula and examined some characteristics of it; we opened the door for a multivariate setting
We present further investigations in the bivariate model: we compute the tail dependence coe cients, we examine the marginal and joint distributions of the componentwise maxima, which leads to an extreme value copula, which – to the best of our knowledge – has not been investigated in the literature yet
Summary
We consider the bivariate lifetime model introduced by Freund [4]. The idea is that the lifetimes of two entities (we refer to them as "institutions") are originally assumed to be Exp(λi) distributed (i = , ), and when one of the entities defaults, it modi es the remaining lifetime of the other entity by increasing the intensity of its original exponential lifetime. We present further investigations in the bivariate model: we compute the tail dependence coe cients, we examine the marginal and joint distributions of the componentwise maxima, which leads to an extreme value copula, which – to the best of our knowledge – has not been investigated in the literature yet.
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