Exchangeable Marshall--Olkin distributions
Source:vignettes/Exchangeable-Marshall-Olkin-distributions.Rmd
Exchangeable-Marshall-Olkin-distributions.RmdThe exchangeable subclass of Marshall–Olkin distribution deserves special consideration. Almost all specific examples of Marshall–Olkin distributions are exchangeable — or are based on mixtures of exchangeable Marshall–Olkin distributions.
For a detailed mathematical treatment of this class, see the references below.
Exchangeability
A random vector is called exchangeable if a reordering of the components does not change its law. An equivalent definition is that the survival function, distribution function, or density (if it exists) does not change if the arguments are permutated.
Condition
A Marshall–Olkin distribution is exchangeable if and only if \[ {\lvert I \rvert} = {\lvert J \rvert} \quad \Rightarrow \quad \lambda_{I} = \lambda_{J} . \]
For this reason, we use the notation \(\lambda_i := \lambda_{I}\), if \(i = {\lvert I \rvert}\).
Reparametrisation
For \(\boldsymbol{t} \geq 0\) with descendingly ordered version \(t_{[1]} \geq \cdots \geq t_{[d]}\) the survival function can be rewritten as
\[ \bar{F}{(\boldsymbol{t})} = \exp{\left \{ - \sum_{i=0}^{d-1}{ a_{i} \cdot t_{[i+1]} } \right \}} , \quad \boldsymbol{t} \geq 0 . \]
The parameters \(a_{0}, a_{1}, \ldots, a_{d-1}\) are defined by
\[ a_{i} = \sum_{j=0}^{d-i-1}{ \binom{d-i-1}{j} \lambda_{j+1} } , \quad i \in {\{ 0, 1, \ldots, d-1 \}} , \]
and the parameters \(\lambda_{1}, \ldots, \lambda_{d}\) can be retrieved by the formula
\[ \lambda_{i} = (-1)^{i-1} \Delta^{i-1} a_{d-i} = \sum_{j=0}^{i-1}{ {(-1)}^{j} \binom{i-1}{j} a_{d-i+j} }, \quad i \in {\{ 1, \ldots, d \}} . \]
A sequences of non-negative parameters \(a_{0} , a_{1}, \ldots, a_{d-1}\) defines a proper Marshall–Olkin distribution if and only if the formula above yields a non-negative sequence \(\lambda_{1}, \ldots, \lambda_{d}\).
Survival copula
For the exchangeable Marshall–Olkin distribution, the survival copula takes the form
\[ \hat{C}{( \boldsymbol{u} )} = u_{(1)} \cdot u_{(2)}^{a_1/a_0} \cdot \ldots \cdot u_{(d)}^{a_{d-1}/a_0} , \quad \boldsymbol{u} \in {[0, 1]}^d \]
where \(u_{(1)} \leq \cdots \leq u_{(d)}\) is the ascendingly ordered version of \(\boldsymbol{u}\).
Fast simulation
Given exchangeable shock-size arrival intensities
\[ \eta_{i} = \binom{d}{i} \lambda_{i} , \quad i \in {\{ 1 , \ldots , d \}} , \]
the corresponding exchangeable Marshall-Olkin distribution can be simulated via its default-counting process \(Z\), which is a Markov process. A sample-path from the default-counting process defines an defines a corresponding order statistic
\[ \tau_{(1)} \leq \cdots \leq \tau_{(d)} \]
We can obtain a random vector with the desired exchangeable Marshall-Olkin distribution by applying an independent random shuffling \(\Pi\):
\[ \tau_ i = \tau_{(\Pi{(i)})} , \quad i \in {\{ 1, \ldots, d \}} . \]