A test for separability in covariance operators of random surfaces
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Date
2017
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Abstract
The assumption of separability is a simplifying and very popular assumption in
the analysis of spatio-temporal or hypersurface data structures. It is often made in
situations where the covariance structure cannot be easily estimated, for example
because of a small sample size or because of computational storage problems. In
this paper we propose a new and very simple test to validate this assumption. Our
approach is based on a measure of separability which is zero in the case of separability
and positive otherwise. The measure can be estimated without calculating
the full non-separable covariance operator. We prove asymptotic normality of the
corresponding statistic with a limiting variance, which can easily be estimated from
the available data. As a consequence quantiles of the standard normal distribution
can be used to obtain critical values and the new test of separability is very easy to
implement. In particular, our approach does neither require projections on subspaces
generated by the eigenfunctions of the covariance operator, nor resampling
procedures to obtain critical values nor distributional assumptions as recently used
by Aston et al. (2017) and Constantinou et al. (2017) to construct tests for separability.
We investigate the finite sample performance by means of a simulation study
and also provide a comparison with the currently available methodology. Finally,
the new procedure is illustrated analyzing wind speed and temperature data.
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Keywords
functional data, surface data structures, space-time processes, separability, minimum distance