Armageddon shock model distributions
Source:vignettes/Armageddon-ESM-distribution.Rmd
Armageddon-ESM-distribution.RmdThe multivariate armageddon shock model family is a very simple example of a mixture of Marshall–Olkin distributions.
For a detailed mathematical treatment of this class, we refer to the references at the end of this document.
Definition and survival function
We say that a random vector has multivariate armageddon shock model distribution if there exist \(\lambda, \lambda^{G} \geq 0\) with \(\lambda + \lambda^{G} > 0\) such that for \(t_{[1]} \geq \cdots \geq t_{[d]}\)
\[ \bar{F}{( \boldsymbol{t} )} = \exp{\left \{ -{[ \lambda + \lambda^{G} ]} t_{[1]} -\lambda \sum_{i = 2}^{d}{ t_{[i]} } \right \}}, \quad \boldsymbol{t} \geq 0 . \]
This random vector has an exchangeable Marshall–Olkin distribution with parameters
\[ \lambda_{I} = \begin{cases} \lambda & {\lvert I \rvert} = 1 , \\ \lambda^{G} & {\lvert I \rvert} = d , \\ 0 & \text{else} . \end{cases} \]
Stochastic model
The random vector can be generated by a classical exogenous shock model, by the Arnold model and its exchangeable version, by a Lévy frailty model with a killed subordinator with drift, and by a mixture of Marshall–Olkin distributions. In the latter case, one combines a vector of independent exponential distributed random variables with rate \(\lambda\) with a comonotone vector with exponential distributed random variables with rate \(\lambda^{G}\) via component-wise minima.