Bootstrap

Abstract

In  Section 10.3 we obtained a central limit theorem for data-based estimators $\widehat{\varvec{\nu }}^*_n(H)$ of cluster functionals ${\varvec{\nu }}^*(H)$. Theorem 10.3.1 allows us to construct confidence intervals for ${\varvec{\nu }}^*(H)$ of the form $\left( \widehat{\varvec{\nu }}^*_{n,{r_{n}}}(H)-\frac{1}{\sqrt{k}}\sigma (H)z_{\beta /2},\widehat{\varvec{\nu }}^*_{n,{r_{n}}}(H) +\frac{1}{\sqrt{k}}\sigma (H)z_{\beta /2}\right) \;,$ where $\sigma ^2(H)$ is the limiting variance of the estimator and $z_{\beta /2}$ is $1-\beta /2$ quantile of the standard normal random variable. Unfortunately, in most cases, the limiting variance depends on unknown parameters and has a complicated form, often an infinite series, as can be seen in the examples of Section 10.4.

12.3 Bibliographical notes

This chapter is essentially an adaptation and extension of [Dre15] and [DDSW18]. The multiplier bootstrap for tail array sums was also considered in [JK19].

The main technical tools come from [Kos03, Kos08] to which we refer for the general theory of various bootstrap methods for i.i.d. sequences. Other bootstrap procedures for tail array sums are considered in [DMC12] and [MZ15].

For Markov chains with a regeneration structure, bootstrap techniques for extremal statistics were developed in [BCT13].