Asyptotic Normality for Maximum Likelihood Estimation and Operational Risk

TL;DR

This paper investigates the validity of asymptotic normality of maximum likelihood estimators in operational risk models, especially with small sample sizes, and assesses the impact on confidence interval accuracy.

Contribution

It provides an analysis of when asymptotic normality holds for common operational risk severity distributions and evaluates the errors in confidence intervals due to its failure.

Findings

Asymptotic normality often fails with small samples in operational risk.

Errors in confidence intervals can be significant when asymptotic assumptions do not hold.

Guidelines for assessing the reliability of MLE-based inference in operational risk.

Abstract

Operational risk models commonly employ maximum likelihood estimation (MLE) to fit loss data to heavy-tailed distributions. Yet several desirable properties of MLE (e.g. asymptotic normality) are generally valid only for large sample-sizes, a situation 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.