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Optimal weighted pooling for inference about the tail index and extreme quantiles

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  • Daouia, Abdelaati
  • Padoan, Simone A.
  • Stupfler, Gilles
Abstract
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 di↵erent 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 e↵ective 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.

Suggested Citation

  • Daouia, Abdelaati & Padoan, Simone A. & Stupfler, Gilles, 2022. "Optimal weighted pooling for inference about the tail index and extreme quantiles," TSE Working Papers 22-1322, Toulouse School of Economics (TSE), revised 07 Jun 2023.
    • Handle: RePEc:tse:wpaper:126783
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    References listed on IDEAS

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    1. Paul Kinsvater & Roland Fried & Jona Lilienthal, 2016. "Regional extreme value index estimation and a test of tail homogeneity," Environmetrics, John Wiley & Sons, Ltd., vol. 27(2), pages 103-115, March.
    2. Rafael Schmidt & Ulrich Stadtmüller, 2006. "Non‐parametric Estimation of Tail Dependence," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 33(2), pages 307-335, June.
    3. A. Dematteo & S. Clémençon, 2016. "On tail index estimation based on multivariate data," Journal of Nonparametric Statistics, Taylor & Francis Journals, vol. 28(1), pages 152-176, March.
    4. Qifa Xu & Chao Cai & Cuixia Jiang & Fang Sun & Xue Huang, 2020. "Block average quantile regression for massive dataset," Statistical Papers, Springer, vol. 61(1), pages 141-165, February.
    5. Haiying Wang & Yanyuan Ma, 2021. "Optimal subsampling for quantile regression in big data," Biometrika, Biometrika Trust, vol. 108(1), pages 99-112.
    Full references (including those not matched with items on IDEAS)

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    More about this item

    Keywords

    Extreme values ; Heavy tails ; Distributed inference ; Pooling ; Testing;
    All these keywords.

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