Open Access
September 2022 Improved inference on risk measures for univariate extremes
Léo R. Belzile, Anthony C. Davison
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Ann. Appl. Stat. 16(3): 1524-1549 (September 2022). DOI: 10.1214/21-AOAS1555


We discuss the use of likelihood asymptotics for inference on risk measures in univariate extreme value problems, focusing on estimation of high quantiles and similar summaries of risk for uncertainty quantification. We study whether higher-order approximation, based on the tangent exponential model, can provide improved inferences. We conclude that inference based on maxima is generally robust to mild model misspecification and that profile likelihood-based confidence intervals will often be adequate, whereas inferences based on threshold exceedances can be badly biased but may be improved by higher-order methods, at least for moderate sample sizes. We use the methods to shed light on catastrophic rainfall in Venezuela, flooding in Venice, and the lifetimes of Italian semisupercentenarians.

Funding Statement

This research was financially supported by the Natural Sciences and Engineering Research Council of Canada and the Swiss National Science Foundation.


We thank the reviewers for particularly useful comments. This work was performed using the R programming language (R Core Team (2021)) with formulae derived through SageMath (The Sage Developers (2021)).


Download Citation

Léo R. Belzile. Anthony C. Davison. "Improved inference on risk measures for univariate extremes." Ann. Appl. Stat. 16 (3) 1524 - 1549, September 2022.


Received: 1 January 2021; Revised: 1 July 2021; Published: September 2022
First available in Project Euclid: 19 July 2022

Digital Object Identifier: 10.1214/21-AOAS1555

Keywords: extreme value distribution , generalized Pareto distribution , higher-order asymptotic inference , Poisson process , profile likelihood , tangent exponential model

Rights: Copyright © 2022 Institute of Mathematical Statistics

Vol.16 • No. 3 • September 2022
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