  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.
