A neural network-based adaptive cut-off approach to normality testing for dependent data

Abstract 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, which typically leads to an inflation in Type I error rates. 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 with adaptive cut-offs by which the Type I error inflation issue is fixed. 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.