Maximum Likelihood Estimation Error and Operational Value-at-Risk Stability

The challenge of using small sample sizes for operational risk capital models fitted via maximum likelihood estimation is well recognized, yet the literature generally provides warning examples rather than a systematic approach. We present a general framework for analyzing maximum likelihood estimation error on operational value-at-risk as a function of sample size for five severity distributions commonly used in operational risk capital models. More specifically, we study the estimation error along three dimensions: the choice of severity distribution, the sample size and the heaviness of the underlying losses. We apply these results to model selection and explore implications for operational risk modeling.