A test for separability in covariance operators of random surfaces

The assumption of separability is a simplifying and very popular assumption in the analysis of spatiotemporal 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. We derive the asymptotic distribution of a corresponding estimate under the null hypothesis and the alternative and develop an asymptotic and a bootstrap test which are very easy to implement. In particular, our approach does neither require projections on subspaces generated by the eigenfunctions of the covariance operator nor distributional assumptions as recently used by (Ann. Statist. 45 (2017) 1431.1461) and (Biometrika 104 425.437) 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 a data example.