Bootstrapping Estimators based on the Block Maxima Method

TL;DR

This paper improves extremal analysis of multivariate time series by proposing a bootstrap method based on circular block maxima, offering consistent inference and comparable variance to sliding block maxima, with practical validation.

Contribution

It introduces a consistent bootstrap approach using circular block maxima for extremal estimators, addressing limitations of naive methods and aligning variance properties with sliding block maxima.

Findings

Circular block bootstrap is consistent for extremal estimators.

Classical bootstrap is consistent for disjoint block maxima.

Methods are validated through Monte Carlo simulations and a precipitation case study.

Abstract

The block maxima method is a standard approach for analyzing the extremal behavior of a potentially multivariate time series. It has recently been found that the classical approach based on disjoint block maxima may be universally improved by considering sliding block maxima instead. However, the asymptotic variance formula for estimators based on sliding block maxima involves an integral over the covariance of a certain family of multivariate extreme value distributions, which makes its estimation, and inference in general, an intricate problem. As an alternative, one may rely on bootstrap approximations: we show that naive block-bootstrap approaches from time series analysis are inconsistent even in i.i.d.\ situations, and provide a consistent alternative based on resampling circular block maxima. As a by-product, we show consistency of the classical resampling bootstrap for disjoint block maxima, and that estimators based on circular block maxima have the same asymptotic variance as their sliding block maxima counterparts. The finite sample properties are illustrated by Monte Carlo experiments, and the methods are demonstrated by a case study of precipitation extremes.