Dual approach to superquantile estimation and applications to density fitting
Sabol, John J., III
Royset, Johannes O.
Buttrey, Samuel E.
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Analysts often concern themselves with the tail regions of distributions, sometimes called extreme events, in order to measure or predict risk. One risk metric, the superquantile, possesses several properties that make it particularly well-suited for risk quantification. Observable data, however, often lack information on extreme events due to various resource constraints, resulting in sample superquantile estimates that often undervalue the true level of risk. By leveraging the dual relationship between superquantiles and superexpectations, we apply constrained optimization on second-order epi-splines to arrive at incrementally better approximations of superquantile values. With these improved estimates, we incorporate additional constraints to improve the fidelity of density estimates in tail regions. We limit our investigation to data with heavy tails, where risk quantification is typically the most difficult. Demonstrations are provided in the form of a known distributional benchmark, historical financial data, and a fluid dynamics model used in the development of a high-speed naval vessel. Results show that accurate quantile and superquantile constraint implementation, in conjunction with empirical statistics and distributional knowledge, can improve tail density estimates by up to 15% for small samples of various heavy-tailed distributions.
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