Estimating operational risk capital
requirements assuming data follows a bi-exponential distribution
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Suppose a risk manager believes that an appropriate model
for a particular type of operational risk exposure involves the loss, ,
coming 50% of the time from an exponential
distribution with parameter and
50% of the time come from an exponential distribution with parameter .
The exponential distribution has
a probability density function, ,
mean, ,
and variance as
follows:
Suppose we want method of moments (MoM) estimators for and
and
we have loss data for
where
the number of losses, ,
is sufficiently large to be able to ignore small sample corrections. Then the
MoM estimators can be derived as follows, where the (sample) moments used in
the estimation are and
.
The pdfs of the individual parts, ,
and of the overall distribution, ,
are:
The mean of is:
where and
By integrating by parts or by noting that for
where and
,
we note that
So method of moments estimators involve (if these
simultaneous equations have a (real) solution):
This is a quadratic equation which has the following
solutions:
The two roots correspond to and
(it
is not possible to differentiate between them given merely the data being
provided). Hence the values of and
are:
In practice it is more likely that the probabilities of
drawing from the underlying exponentials are unknown. This adds an extra degree
of freedom which would introduce the need to include a further (higher) moment
into the parameter estimation process.