Modeling extreme events is one of the central tasks in risk management and planning, as catastrophes and crises put human lives and financial assets at stake. A common approach to estimate the likelihood of extreme events, using extreme value theory (EVT), studies the asymptotic behavior of the ``tail" portion of data, and suggests suitable parametric distributions to fit the data backed up by their limiting behaviors as the data size or the excess threshold grows. We explore an alternate approach to estimate extreme events that is inspired from recent advances in robust optimization. Our approach represents information about tail behaviors as constraints and attempts to estimate a target extremal quantity of interest (e.g, tail probability above a given high level) by imposing an optimization problem to find a conservative estimate subject to the constraints that encode the tail information capturing belief on the tail distributional shape. We first study programs where the feasible region is restricted to distribution functions with convex tail densities, a feature shared by all common parametric tail distributions. We then extend our work by generalizing the feasible region to distribution functions with monotone derivatives and bounded or infinite moments. In both cases, we study the statistical implications of the resulting optimization problems. Through investigating their optimality structures, we also present how the worst-case tail in general behaves as a linear combination of polynomial decay tails. Numerically, we develop results to reduce these optimization problems into tractable forms that allow solution schemes via linear-programming-based techniques.

Modeling of extreme events is a challenging statistical task. Firstly, there is always a limit number of observations and secondly therefore no experience to back test the result. One way of estimating higher quantiles is to fit one of theoretical distributions to the data and extrapolate to the tail. The shortcoming of this approach is that the estimate of the tail is based on the observations in the center of distribution. Alternative approach to this problem is based on idea to split the data into two sub-populations and model body of the distribution separately from the tail. This methodology is applied to non-life insurance losses, where extremes are particularly important for risk management. Never the less, even this approach is not a conclusive solution of heavy tail modeling. In either case, estimated 99.5% percentiles have such high standard errors, that the their reliability is very low. On the other hand this approach is theoretically valid and deserves to be considered as one of the possible methods of extreme value analysis.