Two-sample tests for relevant differences in the eigenfunctions of covariance operators
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Date
2019
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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.
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Keywords
functional data, two-sample tests, self-normalization, relevant tests, functional time series