  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 used by
other available methods of constructing 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.
