Extreme Value Inference for General Heterogeneous Data

We extend extreme value statistics to the general setting of independent data with possibly very different distributions, whereby the extreme value index of the average distribution can be negative, zero, or positive. We present novel asymptotic theory for the moment estimator, 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 i.i.d. 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 and to the tail heaviness of energies of earthquakes around the globe.