| dc.description.abstract | The assumption of separability of the covariance operator
for a random image or hypersurface can be of substantial use in
applications, especially in situations where the accurate estimation
of the full covariance structure is unfeasible, either for computational
reasons, or due to a small sample size. However, inferential tools
to verify this assumption are somewhat lacking in high-dimensional
or functional data analysis settings, where this assumption is most
relevant. We propose here to test separability by focusing on K-
dimensional projections of the difference between the covariance operator
and a nonparametric separable approximation. The subspace
we project onto is one generated by the eigenfunctions of the covariance
operator estimated under the separability hypothesis, negating
the need to ever estimate the full non-separable covariance. We show
that the rescaled difference of the sample covariance operator with its
separable approximation is asymptotically Gaussian. As a by-product
of this result, we derive asymptotically pivotal tests under Gaussian
assumptions, and propose bootstrap methods for approximating the
distribution of the test statistics. We probe the finite sample performance
through simulations studies, and present an application to
log-spectrogram images from a phonetic linguistics dataset. | |
