Two-sample tests for relevant differences in the eigenfunctions of covariance operators

TL;DR

This paper introduces a new relevance-based two-sample testing method for eigenfunctions of covariance operators in functional time series, utilizing self-normalization to improve robustness and applicability in real-world data analysis.

Contribution

It develops a novel relevance testing framework with self-normalized test statistics for functional data, extending existing methods to detect practically meaningful differences.

Findings

The proposed tests perform well in finite samples.

They compare favorably with existing methods.

Application to temperature data demonstrates practical utility.

Abstract

This paper deals with two-sample tests for functional time series data, which have become widely available in conjunction with the advent of modern complex observation systems. Here, particular interest is in evaluating whether two sets of functional time series observations share the shape of their primary modes of variation as encoded by the eigenfunctions of the respective covariance operators. To this end, a novel testing approach is introduced that connects with, and extends, existing literature in two main ways. First, tests are set up in the relevant testing framework, where interest is not in testing an exact null hypothesis but rather in detecting deviations deemed sufficiently relevant, with relevance determined by the practitioner and perhaps guided by domain experts. Second, the proposed test statistics rely on a self-normalization principle that helps to avoid the notoriously difficult task of estimating the long-run covariance structure of the underlying functional time series. The main theoretical result of this paper is the derivation of the large-sample behavior of the proposed test statistics. Empirical evidence, indicating that the proposed procedures work well in finite samples and compare favorably with competing methods, is provided through a simulation study, and an application to annual temperature data.