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### Estimating operational risk capital requirements assuming data follows a bi-exponential distribution

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.