This model represents one type of bivariate extension
of the exponential
distribution that is applicable to certain problems,
in particular,
to two-component systems which can function if one of
the components
has failed. For example, engine failures in
two-engine planes, paired
organs such as peoples' eyes, ears and kidneys.
Suppose \(y_1\) and \(y_2\) are random variables
representing the lifetimes of
two components \(A\) and \(B\)
in a two component system.
The dependence between \(y_1\) and \(y_2\)
is essentially such that the failure of the \(B\) component
changes the parameter of the exponential life distribution
of the \(A\) component from \(\alpha\) to
\(\alpha'\), while the failure of
the \(A\) component
changes the parameter of the exponential life distribution
of the \(B\) component from \(\beta\) to
\(\beta'\).
The joint probability density function is given by
$$f(y_1,y_2) = \alpha \beta' \exp(-\beta' y_2 -
(\alpha+\beta-\beta')y_1) $$
for \(0 < y_1 < y_2\), and
$$f(y_1,y_2) = \beta \alpha' \exp(-\alpha' y_1 -
(\alpha+\beta-\alpha')y_2) $$
for \(0 < y_2 < y_1\).
Here, all four parameters are positive, as well
as the responses
\(y_1\) and \(y_2\).
Under this model, the probability that component \(A\)
is the first to fail is
\(\alpha/(\alpha+\beta)\).
The time to the first failure is distributed as an
exponential distribution with rate
\(\alpha+\beta\). Furthermore, the
distribution of the time from first failure to failure
of the other component is a mixture of
Exponential(\(\alpha'\)) and
Exponential(\(\beta'\)) with proportions
\(\beta/(\alpha+\beta)\)
and \(\alpha/(\alpha+\beta)\)
respectively.
The marginal distributions are, in general, not exponential.
By default, the linear/additive predictors are
\(\eta_1=\log(\alpha)\),
\(\eta_2=\log(\alpha')\),
\(\eta_3=\log(\beta)\),
\(\eta_4=\log(\beta')\).
A special case is when \(\alpha=\alpha'\)
and \(\beta=\beta'\), which means that
\(y_1\) and \(y_2\) are independent, and
both have an ordinary exponential distribution with means
\(1 / \alpha\) and \(1 / \beta\)
respectively.
Fisher scoring is used,
and the initial values correspond to the MLEs of
an intercept model.
Consequently, convergence may take only one iteration.