Extreme value inference for heterogeneous heavy-tailed data: A derandomization theory

A major mathematical difficulty in studying extreme value parameter estimators defined as empirical mean excesses is their reliance on high order statistics above a random threshold. Based on simple yet novel derandomization arguments, we provide sufficient conditions for deriving the joint asymptotic distribution of so-called tail empirical excesses and Expected Shortfall with the underlying threshold level. This high-level result allows for a strong degree of heterogeneity in the data-generating process as well as serial dependence. When the observations are independent and their average distribution is heavy-tailed, we obtain asymptotic normality results for the Hill estimator of the extreme value index, the Weissman estimator of extreme quantiles, and two estimators of Expected Shortfall above an extreme level, under substantially weaker, yet easily verifiable and interpretable conditions than those prevailing in the recent literature. In particular, we establish precise closed-form expressions for the asymptotic bias and variance of each estimator. Our assumptions hold in a wide range of models where existing results may not apply, including scenarios of contaminated samples, pooled samples from several populations, heterogeneous location-scale models and the situation where observed covariate information is ignored. We discuss practical consequences of our results on simulated data and two real data applications to cyber risk and financial risk management.