AbstractsStatistics

This dissertation consists of results in estimation, prediction and bounding risk for multivariate extremes. Regarding estimation, we establish a consistent and asymptotically normal M-estimator that is applicable to a wide variety of max-stable models, i.e. the class of distributions arising as the limit of component-wise maxima. Such processes play a fundamental role in modeling extreme phenomena, but are challenging to work with due to a lack of tractable likelihoods. Our method circumvents intractable likelihoods, working directly with distribution functions of max-stable processes which are readily available or can be approximated in a precise manner. Our second contribution is in the area of prediction for spatial extremes, specifically extreme precipitation. We introduce Gauss-Pareto random fields as a flexible class of models that capture essential non-trivial extremal dependence characteristics, yet remain amenable to standard Bayesian MCMC techniques. We apply Gauss-Pareto processes to spatial prediction of extreme precipitation over Sweden and show that Gauss-Pareto models yield skillful predictions in practice. Lastly we establish universal bounds on the extreme Value-at-Risk of various functionals of portfolio losses. Specifically, the maximum portfolio loss and the sum of tail dependent losses under given summary measures of tail dependence called extremal coefficients. While extremal coefficients are finite dimensional and consistent estimators are readily obtainable, they do not fully characterize tail dependence. Prior to this work, it was not known how extremal coefficients constrain Value-at-Risk for extreme losses. The solution involves solving an optimization problem over an infinite dimensional space of measures. Here we prove that the optimization problem can be reduced to a convex optimization problem in finite dimensions and develop algorithms to compute the bounds.