On Certain Goodness-of-Fit Tests in Operational Risk Modeling

Measurement of operational risk, through a loss distribution approach (LDA), for the purpose of bank capitalization poses significant modeling challenges. As part of LDA, the severity of losses characterizing the monetary impact of potential operational risk events has to be modelled via a severity distribution. The selection of a best-fit severity distribution is essential for the accurate modeling of the impacts. In this article, we provide an analysis of distributional properties of a family of goodness-of-fit tests suitable for more accurate selection of best-fit severity distributions. We describe first some classical results which are not known widely. We also demonstrate that certain goodness-of-fit test statistics popular in the financial industry do not have limiting distributions. For this reason, we provide a normalization that leads to a nondegenerate asymptotic distribution. Finally, a number of auxiliary results are presented that are of independent interest