Date of Award

May 2016

Degree Type

Dissertation

Degree Name

Doctor of Philosophy

Department

Mathematics

First Advisor

Vytaras Brazauskas

Committee Members

Jay Beder, Jugal Ghorai, Hans Volkmer, Wei Wei

Abstract

In this dissertation, we focus on statistical aspects of operational risk modeling. Specifically, we are interested in understanding the effects of model uncertainty on capital reserves due to data truncation and in developing better model selection tools for truncated and shifted parametric distributions. We first investigate the model uncertainty question which has been unanswered for many years because researchers, practitioners, and regulators could not agree on how to treat the data collection threshold in operational risk modeling. There are several approaches under consideration—the empirical approach, the “naive” approach, the shifted approach, and the truncated approach—for fitting the loss severity distribution. Since each approach is based on a different set of assumptions, different probability models emerge. Thus, model uncertainty arises. When possible we investigate such model uncertainty analytically using asymptotic theorems of mathematical statistics and several parametric distributions commonly used for operational risk modeling, otherwise we rely on Monte Carlo simulations. The effect of model uncertainty on risk measurements is quantified by evaluating the probability of each approach producing conservative capital allocations based on the value-at-risk measure. These explorations are further illustrated using a real data set for legal losses in a business unit. After clarifying some prevailing misconceptions around the model uncertainty issue in operational risk modeling, we then employ standard (Akaike Information Criterion, AIC, and Bayesian Information Criterion, BIC) and introduce new model selection tools for truncated and shifted parametric models. We find that the new criteria, which are based on information complexity and asymptotic mean curvature of the model likelihood, are more effective at distinguishing between the competing models than AIC and BIC.

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