Moving Block and Tapered Block Bootstrap for Functional Time Series with an Application to the K-Sample Mean Problem

TL;DR

This paper develops and validates moving block and tapered block bootstrap methods for dependent functional time series, enabling reliable inference on mean functions and covariance operators in infinite-dimensional spaces.

Contribution

It introduces new bootstrap procedures for functional time series, proving their asymptotic validity and applying them to test equality of mean functions across multiple series.

Findings

Bootstrap methods are consistent for covariance estimation.

Valid for testing mean function equality under dependence.

Finite sample performance is demonstrated through simulations.

Abstract

We consider infinite-dimensional Hilbert space-valued random variables that are assumed to be temporal dependent in a broad sense. We prove a central limit theorem for the moving block bootstrap and for the tapered block bootstrap, and show that these block bootstrap procedures also provide consistent estimators of the long run covariance operator. Furthermore, we consider block bootstrap-based procedures for fully functional testing of the equality of mean functions between several independent functional time series. We establish validity of the block bootstrap methods in approximating the distribution of the statistic of interest under the null and show consistency of the block bootstrap-based tests under the alternative. The finite sample behaviour of the procedures is investigated by means of simulations. An application to a real-life dataset is also discussed.