On maximum likelihood estimation of operational loss distributions

This paper develops a likelihood-based methodology to estimate loss distributions and compute Capital at Risk in risk management applications. In particular, we deal with the problem of estimating severity distributions with censored and truncated operational losses, for which numerical maximization of the likelihood function by means of standard optimization tools may be difficult. We show that, under the standard hypothesis of lognormal severity, maximum likelihood estimation can be performed by means of the EM algorithm. We derive the relevant equations of the algorithm and apply it to operational loss data. Finally, a simulation study shows that, in this setup, the EM algorithm has more desirable properties than both the BFGS algorithm and the Nelder-Mead simplex algorithm.