  This paper investigates pooling strategies for tail index and extreme
quantile estimation from heavy-tailed data. To fully exploit the information
contained in several samples, we present general weighted pooled Hill
estimators of the tail index and weighted pooled Weissman estimators of extreme
quantiles calculated through a nonstandard geometric averaging scheme. We
develop their large-sample asymptotic theory across a fixed number of samples,
covering the general framework of heterogeneous sample sizes with different and
asymptotically dependent distributions. Our results include optimal choices of
pooling weights based on asymptotic variance and MSE minimization. In the
important application of distributed inference, we prove that the
variance-optimal distributed estimators are asymptotically equivalent to the
benchmark Hill and Weissman estimators based on the unfeasible combination of
subsamples, while the AMSE-optimal distributed estimators enjoy a smaller AMSE
than the benchmarks in the case of large bias. We consider additional scenarios
where the number of subsamples grows with the total sample size and effective
subsample sizes can be low. We extend our methodology to handle serial
dependence and the presence of covariates. Simulations confirm that our pooled
estimators perform virtually as well as the benchmark estimators. Two
applications to real weather and insurance data are showcased.
