EVT-Based Rate-Preserving Distributional Robustness for Tail Risk Functionals

arXiv:2506.16230v2 Announce Type: replace-cross
Abstract: Risk measures such as Conditional Value-at-Risk (CVaR) focus on extreme losses, where scarce tail data makes model error unavoidable. To hedge misspecification, one evaluates worst-case tail risk over an ambiguity set. Using Extreme Value Theory (EVT), we derive first-order asymptotics for worst-case tail risk for a broad class of tail-risk measures under standard ambiguity sets, including Wasserstein balls and $phi$-divergence neighborhoods. We show that robustification can alter the nominal tail asymptotic scaling as the tail level $betato0$, leading to excess risk inflation. Motivated by this diagnostic, we propose a tail-calibrated ambiguity design that preserves the nominal tail asymptotic scaling while still guarding against misspecification. Under standard domain of attraction assumptions, we prove that the resulting worst-case risk preserves the baseline first-order scaling as $betato0$, uniformly over key tuning parameters, and that a plug-in implementation based on consistent tail-index estimation inherits these guarantees. Synthetic and real-data experiments show that the proposed design avoids the severe inflation often induced by standard ambiguity sets.