May 2024 Optimal weighted pooling for inference about the tail index and extreme quantiles
Abdelaati Daouia, Simone A. Padoan, Gilles Stupfler
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Bernoulli 30(2): 1287-1312 (May 2024). DOI: 10.3150/23-BEJ1632


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 the statistical inferential theory of our pooled estimators. Two applications to real weather and insurance data are showcased.


The authors acknowledge an anonymous Associate Editor and two anonymous reviewers for their helpful comments that led to an improved version of this paper. A. Daouia and G. Stupfler acknowledge financial support from the French National Research Agency under the grants ANR-19-CE40-0013 (ExtremReg project) and ANR-17-EURE-0010 (EUR CHESS), as well as from the TSE-HEC ACPR Chair and from an AXA Research Fund Award on “Mitigating risk in the wake of the COVID-19 pandemic”. G. Stupfler acknowledges further support from the Centre Henri Lebesgue (ANR-11-LABX-0020-01). S.A. Padoan is supported by the Bocconi Institute for Data Science and Analytics (BIDSA), Italy.


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Abdelaati Daouia. Simone A. Padoan. Gilles Stupfler. "Optimal weighted pooling for inference about the tail index and extreme quantiles." Bernoulli 30 (2) 1287 - 1312, May 2024.


Received: 1 April 2022; Published: May 2024
First available in Project Euclid: 31 January 2024

MathSciNet: MR4699553
Digital Object Identifier: 10.3150/23-BEJ1632

Keywords: Extreme values , heavy tails , inference , pooling , testing


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Vol.30 • No. 2 • May 2024
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