# Testing Equality of Autocovariance Operators for Functional Time Series

## TL;DR

This paper introduces a bootstrap-based test for equality of autocovariance operators in functional time series, providing theoretical validation, simulation results, and real data application.

## Contribution

It proposes a novel moving block bootstrap method for testing autocovariance operator equality, overcoming common bootstrap challenges in functional data analysis.

## Key findings

- Validates the bootstrap test under the null hypothesis.
- Shows consistency of the test under the alternative.
- Demonstrates good finite sample performance through simulations.

## Abstract

We consider strictly stationary stochastic processes of Hilbert space-valued random variables and focus on fully functional tests for the equality of the lag-zero autocovariance operators of several independent functional time series. A moving block bootstrap-based testing procedure is proposed which generates pseudo random elements that satisfy the null hypothesis of interest. It is based on directly bootstrapping the time series of tensor products which overcomes some common difficulties associated with applications of the bootstrap to related testing problems. The suggested methodology can be potentially applied to a broad range of test statistics of the hypotheses of interest. As an example, we establish validity for approximating the distribution under the null of a test statistic based on the Hilbert-Schmidt distance of the corresponding sample lag-zero autocovariance operators, and show consistency under the alternative. As a prerequisite, we prove a central limit theorem for the moving block bootstrap procedure applied to the sample autocovariance operator which is of interest on its own. The finite sample size and power performance of the suggested moving block bootstrap-based testing procedure is illustrated through simulations and an application to a real-life dataset is discussed.