Tail Analysis without Parametric Models: A Worst-case Perspective

TL;DR

This paper introduces a non-parametric method for tail analysis that computes worst-case bounds based on tail convexity, avoiding the pitfalls of parametric assumptions and providing more reliable extremal performance estimates.

Contribution

It proposes a novel approach to tail analysis using worst-case bounds under tail convexity, eliminating the need for parametric distribution fitting.

Findings

The worst-case tail behavior is either extremely light or heavy.

Low-dimensional nonlinear programs effectively distinguish tail types.

The method offers more reliable performance bounds than traditional parametric methods.

Abstract

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.