Selection and Application of Distorted Risk Measures
This thesis develops and illustrates a methodology for the selection of probability distributions and distortion functions associated with risk scenarios resulting from military capability shortfalls. Distorted (or transformed) risk measures are analyzed and applied to account for loss scenarios that may occur with low frequency but result in catastrophic outcomes. After reviewing the rudimentary concepts of distortion, four wellknown continuous distributions, suitable for modeling risk scenarios, are chosen using defined criteria. Based on subject matter expert inputs, a simple method for assigning exactly one of the four distributions to any risk scenario is proposed. Four parametric distortion functions from the finance and insurance literature are then selected and applied to each of the chosen distributions. The distortion effects are examined analytically, graphically, and empirically, and broad-based recommendations are recorded as to the instances when one distortion function might be preferred over others. An end-to-end notional problem -- in which a subset of available mitigation measures are selected to counteract a multi-faceted risk environment -- illustrates the means by which the proposed methodology may be used to affect future systems acquisition through the Capabilities Review and Risk Assessment (CRRA) process of the United States Air Force.
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