A subclass of exchangeable Marshall–Olkin distributions can be sampled with a so-called Lévy frailty model. This alternative stochastic model is very efficient in high dimensions.
For a detailed mathematical treatment of this class, we refer to the references at the end of this document.
The extendible subclass
An exchangeable Marshall–Olkin distribution with parameters \(a_{0}, \ldots, a_{d-1}\) is called extendible if there exists a sequence \(\tau_{1}, \tau_{2}, \ldots\) such that each finite margin has a Marshall–Olkin distribution and
\[ \mathbb{P}{\left( \tau_{1} > t_{1} , \ldots \tau_{d} > t_{d} \right)} = \exp{\left \{ -\sum_{i = 0}^{d-1}{ a_{i} t_{[i+1]} } \right \}}, \quad \boldsymbol{t} \geq 0 . \]
The Lévy frailty representation
Each extendible Marshall–Olkin distribution can be uniquely linked (in law) to a Lévy subordinator \(\Lambda\) and a iid unit exponential random variables \(E_{1}, \ldots , E_{d}\) such that \(\boldsymbol{\tau}\) has this distribution with
\[ \tau_{i} = \inf{\left \{ t > 0 \ : \ \Lambda_{t} \geq E_i \right \}}, \quad i \in {\{ 1, \ldots, d \}} . \]
If \(\psi\) is the Laplace exponent of the Lévy subordinator, i.e., \(\psi{(x)} = -\log\mathbb{E}{[\exp{\{ - x \Lambda_1\}}]}\), then
\[ a_{i} = \psi{(i+1)} - \psi{(i)} , \quad i \in {\{ 1, \ldots, d \}} . \]
Homogeneous Poisson process
A simple example is the case, when \(\Lambda\) is a homogenous Poisson process with rate \(\lambda > 0\). In this case
\[ \psi{(x)} = \lambda {\left( 1 - \exp{\{ -x \}} \right)} , \quad x \geq 0 . \]
Exponential killing and drift
Another simple example is the case, when the subordinator has a deterministic drift \(b \geq 0\) and is killed (send to infinity) at a certain rate \(a \geq 0\). That means
\[ \Lambda_{t} = N_{t}^{G} \cdot \infty + b t , \quad t \geq 0 , \]
where \(N^{G}\) is a homogeneous Poisson process with rate \(a\).
We have
\[ \psi{(x)} = a 1_{(0, \infty)}{(x)} + b x , \quad x \geq 0 . \]
This model produces a multivariate armageddon shock model distribution.
Compound Poisson process
The class of compound Poisson subordinators is the largest class of Lévy subordinators which can be used to sample exactly from a Marshall–Olkin distribution. Here, the Lévy subordinator has the form
\[ \Lambda_{t} = N_{t}^{G} \cdot \infty + b t + \sum_{i = 1}^{N_{t}} X_{i} , \quad t \geq 0 , \]
where \(N^{G}\) is a homogeneous Poisson process with rate \(a \geq 0\), \(b \geq 0\) is a deterministic drift, \(N\) is a homogeneous Poisson process with rate \(\lambda \geq 0\), and \({\{ X_{i} \}}_{i \in \mathbb{N}}\) is an iid sequence of non-negative random variables (all objects are independent).
We have
\[ \psi{(x)} = a 1_{(0, \infty)}{(x)} + b x + \lambda {\left( 1 - \mathcal{L}{(X_{1}; x)} \right)} , \quad x \geq 0 , \]
where \(x \mapsto \mathcal{L}{(X_{1}, x)}\) is the Laplace transform of \(X_{1}\).