On a family of weighted Cramér–von Mises goodness-of-fit tests in operational risk modeling

The measurement of operational risk via the loss distribution approach (LDA) for bank capitalization purposes offers significant modeling challenges. Under the LDA, the severity of losses characterizing the monetary impact of potential operational risk events is modeled via a severity distribution. The selection of best-fit severity distributions that properly capture tail behavior is essential for accurate modeling. In this paper, we analyze the limiting properties of a family of weighted Cramér–von Mises (WCvM) goodness-of-fit test statistics, with weight function ψ⠢(t)=1/(1-t)β, which are suitable for more accurately selecting severity distributions. Specifically, we apply classical theory to determine if limiting distributions exist for these WCvM test statistics under a simple null hypothesis. We show that limiting distributions do not exist for β≥2. For β=2, we provide a normalization that leads to a nondegenerate limiting distribution. Where limiting distributions originally exist for β<2 or are obtained through normalization, we show that, for 1.5≤β≤2, the tests’ practical utility may be limited due to a very slow convergence of the finite-sample distribution to the asymptotic regime. Our results suggest that the tests provide greater utility when β<1.5, and that utility is questionable for β≥1.5, as only Monte Carlo schemes are practical in this case, even for very large samples.