Extreme Value Statistics for General Heterogeneous Data through the Average Tail

原文摘要

In the general setting of independent data with possibly very different distributions, extreme value estimators inevitably target the tail of the average distribution function. We consider all possible cases, that is, the extreme value index of the average distribution can be negative, zero, or positive, and we present novel asymptotic theory for the moment estimator. Our results require a different and much more challenging proof than those for the power-law case and are based on a uniform central limit theorem for the underlying weighted tail empirical process. We find that, due to the heterogeneity of the data, the asymptotic variance of the moment estimator can be much smaller than that in the iid case. We also unravel the improved performance of high quantile and endpoint estimators in this setup. In case of a heavy tail, we ameliorate the Hill estimator by taking an optimal combination of the Hill and the moment estimator. Simulations show the good finite-sample behavior of our limit results. Finally we present applications to the maximum lifespan of monozygotic twins, the ultimate 200m running world records, and to the tail heaviness of energies of earthquakes around the globe. Supplementary materials for this article are available online, including a standardized description of the materials available for reproducing the work.

摘要来源：OpenAlex