Marshall–Olkin exponential distribution

In applied statistics, the Marshall–Olkin exponential distribution is any member of a certain family of continuous multivariate probability distributions with positive-valued components. It was introduced by Albert W. Marshall and Ingram Olkin.^[1] One of its main uses is in reliability theory, where the Marshall–Olkin copula models the dependence between random variables subjected to external shocks. ^{[2] [3] [4]}

Marshall–Olkin exponential

Support	${\displaystyle x\in [0,\infty )^{b}}$

Definition

Let ${\displaystyle \{E_{B}:\varnothing \neq B\subset \{1,2,\ldots ,b\}\}}$ be a set of independent, exponentially distributed random variables, where ${\displaystyle E_{B}}$ has mean ${\displaystyle 1/\lambda _{B}}$. Let

${\displaystyle T_{j}=\min\{E_{B}:j\in B\},\ \ j=1,\ldots ,b.}$

The joint distribution of ${\displaystyle T=(T_{1},\ldots ,T_{b})}$ is called the Marshall–Olkin exponential distribution with parameters ${\displaystyle \{\lambda _{B},B\subset \{1,2,\ldots ,b\}\}.}$

Concrete example

Suppose b = 3. Then there are seven nonempty subsets of { 1, ..., b } = { 1, 2, 3 }; hence seven different exponential random variables:

${\displaystyle E_{\{1\}},E_{\{2\}},E_{\{3\}},E_{\{1,2\}},E_{\{1,3\}},E_{\{2,3\}},E_{\{1,2,3\}}}$

Then we have:

${\displaystyle {\begin{aligned}T_{1}&=\min\{E_{\{1\}},E_{\{1,2\}},E_{\{1,3\}},E_{\{1,2,3\}}\}\\T_{2}&=\min\{E_{\{2\}},E_{\{1,2\}},E_{\{2,3\}},E_{\{1,2,3\}}\}\\T_{3}&=\min\{E_{\{3\}},E_{\{1,3\}},E_{\{2,3\}},E_{\{1,2,3\}}\}\\\end{aligned}}}$