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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.