Journal of Operational Risk

Operational risk models and asymptotic normality of maximum likelihood estimation

Paul Larsen

  • Asymptotic normality assumptions are usually not verifiable for severity distributions
  • Graphical and numerical of asymptotic normality vary widely on these distributions
  • The normal approximation for parameter confidence intervals performs relatively well


Operational risk models commonly employ maximum likelihood estimation (MLE) to fit loss data to heavy-tailed distributions. Yet several desirable properties of MLE (eg, asymptotic normality) are generally valid only for large sample sizes, a situation that is rarely encountered in operational risk. In this paper, we study how asymptotic normality does, or does not, hold for common severity distributions in operational risk models. We then apply these results to evaluate errors caused by failure of asymptotic normality in constructing confidence intervals around the MLE fitted parameters.

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