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February 2021 Multiple block sizes and overlapping blocks for multivariate time series extremes
Nan Zou, Stanislav Volgushev, Axel Bücher
Ann. Statist. 49(1): 295-320 (February 2021). DOI: 10.1214/20-AOS1957


Block maxima methods constitute a fundamental part of the statistical toolbox in extreme value analysis. However, most of the corresponding theory is derived under the simplifying assumption that block maxima are independent observations from a genuine extreme value distribution. In practice, however, block sizes are finite and observations from different blocks are dependent. Theory respecting the latter complications is not well developed, and, in the multivariate case, has only recently been established for disjoint blocks of a single block size. We show that using overlapping blocks instead of disjoint blocks leads to a uniform improvement in the asymptotic variance of the multivariate empirical distribution function of rescaled block maxima and any smooth functionals thereof (such as the empirical copula), without any sacrifice in the asymptotic bias. We further derive functional central limit theorems for multivariate empirical distribution functions and empirical copulas that are uniform in the block size parameter, which seems to be the first result of this kind for estimators based on block maxima in general. The theory allows for various aggregation schemes over multiple block sizes, leading to substantial improvements over the single block length case and opens the door to further methodology developments. In particular, we consider bias correction procedures that can improve the convergence rates of extreme-value estimators and shed some new light on estimation of the second-order parameter when the main purpose is bias correction.


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Nan Zou. Stanislav Volgushev. Axel Bücher. "Multiple block sizes and overlapping blocks for multivariate time series extremes." Ann. Statist. 49 (1) 295 - 320, February 2021.


Received: 1 March 2019; Revised: 1 February 2020; Published: February 2021
First available in Project Euclid: 29 January 2021

Digital Object Identifier: 10.1214/20-AOS1957

Primary: 62E20, 62G32
Secondary: 60G70, 62G20

Rights: Copyright © 2021 Institute of Mathematical Statistics


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Vol.49 • No. 1 • February 2021
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