  There is a wide availability of methods for testing normality under the
assumption of independent and identically distributed data. When data are
dependent in space and/or time, however, assessing and testing the marginal
behavior is considerably more challenging, as the marginal behavior is impacted
by the degree of dependence. We propose a new approach to assess normality for
dependent data by non-linearly incorporating existing statistics from normality
tests as well as sample moments such as skewness and kurtosis through a neural
network. We calibrate (deep) neural networks by simulated normal and non-normal
data with a wide range of dependence structures and we determine the
probability of rejecting the null hypothesis. We compare several approaches for
normality tests and demonstrate the superiority of our method in terms of
statistical power through an extensive simulation study. A real world
application to global temperature data further demonstrates how the degree of
spatio-temporal aggregation affects the marginal normality in the data.
