  In materials science and many other application domains, 3D information can
often only be extrapolated by taking 2D slices. In topological data analysis,
persistence vineyards have emerged as a powerful tool to take into account
topological features stretching over several slices. In the present paper, we
illustrate how persistence vineyards can be used to design rigorous statistical
hypothesis tests for 3D microstructure models based on data from 2D slices.
More precisely, by establishing the asymptotic normality of suitable
longitudinal and cross-sectional summary statistics, we devise goodness-of-fit
tests that become asymptotically exact in large sampling windows. We illustrate
the testing methodology through a detailed simulation study and provide a
prototypical example from materials science.
