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### Estimating operational risk capital requirements assuming data follows a bi-uniform or a triangular distribution (using maximum likelihood)

Suppose a risk manager believes that an appropriate model for a particular type of operational risk exposure involves the loss, , never exceeding an upper limit, , and the probability density function  taking the form of a bi-uniform distribution:

where  are all constant.

Suppose we want to estimate maximum likelihood estimators for ,  and  given losses of , say and hence to estimate a Value-at-Risk for a given confidence level for this loss type, assuming that the probability distribution has the form set out above.

We note that  for  to correspond to a probability density function, so:

Suppose the  losses,  , are assumed to be independent draws from a distribution with probability density function  and suppose  of these losses are less  and  are greater than . The likelihood is then:

This will be maximised for some value that has , i.e. has  at least as large as . In such circumstances the likelihood is maximised when the log likelihood is maximised which will be when , i.e. when , i.e. when:

(assuming  and )

For these values of  and  the log likelihood is then:

In most circumstances this will be maximised when  is as small as possible, provided  is still at least as large as  so the maximum likelihood estimators are:

However, it is occasionally necessary to consider the case where we select a  and have  and/or  equal to zero.

To estimate a VaR at a confidence level  we need to find the value  for which the loss is expected to exceed  only % of the time, i.e.  such that (if ):

A similar approach can be used for tri-uniform distributions or other more complex piecewise uniform distributions.

Somewhat more complex is if  takes the form of a triangular distribution, i.e. where:

where  are all constant.

Nguyen and McLachlan (2016), when developing a new algorithm for maximum likelihood estimation of triangular or more general polygonal distributions, note that in the case where  and  then Oliver (1972), “A maximum likelihood oddity”, American Statistician 26, 43–44, indicates that  for some  and moreover that if the sample is ordered so that  then  where:

Asymptotically, on average, the number of observations in  appears to be approximately two.

We can apply this reasoning to the more general triangular distribution where we do not know  by again noting that in many cases the log-likelihood will be minimised by setting  to be as small as possible, provided  is still at least as large as , and then separately handling cases where the likelihood can be improved by selecting a higher value for .