A test for separability in covariance operators of random surfaces

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.