  A common bottleneck in evaluating extremal performance measures is that, due
to their very nature, tail data are often very limited. The conventional
approach selects the best probability distribution from tail data using
parametric fitting, but the validity of the parametric choice can be difficult
to verify. This paper describes an alternative based on the computation of
worst-case bounds under the geometric premise of tail convexity, a feature
shared by all common parametric tail distributions. We characterize the
optimality structure of the resulting optimization problem, and demonstrate
that the worst-case convex tail behavior is in a sense either extremely
light-tailed or extremely heavy-tailed. We develop low-dimensional nonlinear
programs that distinguish between the two cases and compute the worst-case
bound. We numerically illustrate how the proposed approach can give more
reliable performances than conventional parametric methods.
