Multiple block sizes and overlapping blocks for multivariate time series extremes

TL;DR

This paper demonstrates that overlapping blocks improve the efficiency of multivariate extreme value estimators and develops new theoretical results for their asymptotic behavior, enabling better bias correction and multiple block size aggregation.

Contribution

It introduces the use of overlapping blocks for multivariate extremes, providing new asymptotic variance results and functional limit theorems that enhance estimation accuracy.

Findings

Overlapping blocks reduce asymptotic variance of estimators.

Functional central limit theorems are established for block maxima.

Aggregation over multiple block sizes improves estimation performance.

Abstract

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.