Detecting relevant differences in the covariance operators of functional time series -- a sup-norm approach

TL;DR

This paper introduces bootstrap-based statistical tests for detecting small differences or changes in the covariance operators of functional time series, focusing on the sup-norm measure rather than exact equality.

Contribution

It develops novel inference tools for the covariance operators of functional time series, emphasizing small deviations and change points using a sup-norm approach.

Findings

Bootstrap tests effectively detect small differences in covariance operators.

Asymptotic properties of the tests are established.

Simulation studies demonstrate good finite sample performance.

Abstract

In this paper we propose statistical inference tools for the covariance operators of functional time series in the two sample and change point problem. In contrast to most of the literature the focus of our approach is not testing the null hypothesis of exact equality of the covariance operators. Instead we propose to formulate the null hypotheses in them form that "the distance between the operators is small", where we measure deviations by the sup-norm. We provide powerful bootstrap tests for these type of hypotheses, investigate their asymptotic properties and study their finite sample properties by means of a simulation study.