/

### 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 .