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This paper investigates pooling strategies for tail index and extremequantile estimation from heavy-tailed data. To fully exploit the informationcontained in several samples, we present general weighted pooled Hillestimators of the tail index and weighted pooled Weissman estimators of extremequantiles calculated through a nonstandard geometric averaging scheme. Wedevelop their large-sample asymptotic theory across a fixed number of samples,covering the general framework of heterogeneous sample sizes with different andasymptotically dependent distributions. Our results include optimal choices ofpooling weights based on asymptotic variance and MSE minimization. In theimportant application of distributed inference, we prove that thevariance-optimal distributed estimators are asymptotically equivalent to thebenchmark Hill and Weissman estimators based on the unfeasible combination ofsubsamples, while the AMSE-optimal distributed estimators enjoy a smaller AMSEthan the benchmarks in the case of large bias. We consider additional scenarioswhere the number of subsamples grows with the total sample size and effectivesubsample sizes can be low. We extend our methodology to handle serialdependence and the presence of covariates. Simulations confirm that our pooledestimators perform virtually as well as the benchmark estimators. Twoapplications to real weather and insurance data are showcased.

Optimal pooling and distributed inference for the tail index and extreme quantiles