On a Family of Weighted Cramér–Von Mises Goodness-of-Fit Tests in Operational Risk Modeling

21 Pages Posted: 9 Jun 2017

Date Written: June 8, 2017


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.

Keywords: operational risk, hypothesis testing, goodness-of-fit, limiting distributions, convergence

Suggested Citation

Mayorov, Kirill and Hristoskov, James and Balakrishnan, Narayanaswamy, On a Family of Weighted Cramér–Von Mises Goodness-of-Fit Tests in Operational Risk Modeling (June 8, 2017). Journal of Operational Risk, Forthcoming. Available at SSRN: https://ssrn.com/abstract=2983048

Kirill Mayorov (Contact Author)

RBC Financial Group ( email )

155 Wellington St. West, 16-th floor
Toronto, ON M5V 3K7

James Hristoskov

RBC Financial Group ( email )

155 Wellington Ave. West
Toronto, Ontario M5V3K7

Narayanaswamy Balakrishnan

McMaster University ( email )

1280 Main Street West
Hamilton, Ontario L8S 4M4

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