Bootstrapping Hill estimator and tail array sums for regularly varying time series

Abstract

In the extreme value analysis of stationary regularly varying time series, tail array sums form a broad class of statistics suitable to analyze their extremal behavior. This class includes for example, the Hill estimator or estimators of the extremogram and the tail dependence coefficient.

General asymptotic theory for tail array sums has been developed by Rootzén et al. (Ann. Appl. Probab. 8 (1998) 868–885) under mixing conditions and in Kulik et al. (Stochastic Process. Appl. 129 (2019) 4209–4238) for functions of geometrically ergodic Markov chains. A more general framework of cluster functionals is presented in Drees and Rootzén (Ann. Statist. 38 (2010) 2145–2186).

However, the resulting limiting distributions turn out to be very complex and cumbersome to estimate as they usually depend on the whole extremal dependence structure of the time series. Hence, a suitable bootstrap procedure is desired, but available bootstrap consistency results for tail array sums are scarce. In this paper, following Drees (Drees (2015)), we consider a multiplier block bootstrap to estimate the limiting distribution of tail array sums. We prove that, conditionally on the data, an appropriately constructed multiplier block bootstrap statistic converges to the correct limiting distribution. Interestingly, in contrast, it turns out that an apparently natural, but naïve application of the multiplier block bootstrap scheme does not yield the correct limit.

In simulations, we provide numerical evidence of our theoretical findings and illustrate the superiority of the proposed multiplier block bootstrap over some obvious competitors. The proposed bootstrap scheme proves to be computationally efficient in comparison to other approaches.