Two-Sample Tests for Relevant Differences in the Eigenfunctions of Covariance Operators

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