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.

翻译：现有大量方法可用于独立同分布数据下的正态性检验。然而，当数据在空间和/或时间上存在相依性时，评估和检验边际行为变得相当具有挑战性，因为边际行为会受到依赖程度的影响。我们提出了一种新方法，通过神经网络将正态性检验中的现有统计量以及样本矩（如偏度和峰度）进行非线性整合，从而评估相依数据的正态性。我们利用具有广泛依赖结构的模拟正态与非正态数据对（深度）神经网络进行校准，并确定拒绝原假设的概率。我们比较了多种正态性检验方法，并通过广泛的模拟研究证明了我们的方法在统计功效方面的优越性。对全球温度数据的实际应用进一步展示了时空聚合程度如何影响数据的边际正态性。