The assumption of normality has underlain much of the development of statistics, including spatial statistics, and many tests have been proposed. In this work, we focus on the multivariate setting and first review the recent advances in multivariate normality tests for i.i.d. data, with emphasis on the skewness and kurtosis approaches. We show through simulation studies that some of these tests cannot be used directly for testing normality of spatial data. We further review briefly the few existing univariate tests under dependence (time or space), and then propose a new multivariate normality test for spatial data by accounting for the spatial dependence. The new test utilises the union-intersection principle to decompose the null hypothesis into intersections of univariate normality hypotheses for projection data, and it rejects the multivariate normality if any individual hypothesis is rejected. The individual hypotheses for univariate normality are conducted using a Jarque–Bera type test statistic that accounts for the spatial dependence in the data. We also show in simulation studies that the new test has a good control of the type I error and a high empirical power, especially for large sample sizes. We further illustrate our test on bivariate wind data over the Arabian Peninsula.

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你们都正常吗？这取决于！

正态性假设是包括空间统计在内的大部分统计发展的基础，并且已经提出了许多检验。在这项工作中，我们关注多变量设置，并首先回顾 iid 数据多变量正态性检验的最新进展，重点是偏度和峰态方法。我们通过模拟研究表明，其中一些测试不能直接用于测试空间数据的正态性。我们进一步简要回顾了几个现有的依赖（时间或空间）下的单变量测试，然后通过考虑空间依赖性提出了一种新的空间数据多变量正态性测试。新测试利用并交原则将零假设分解为投影数据的单变量正态假设的交集，如果拒绝任何单个假设，它就会拒绝多元正态性。单变量正态性的各个假设是使用 Jarque-Bera 类型检验统计量进行的，该检验统计量解释了数据中的空间依赖性。我们还在模拟研究中表明，新测试可以很好地控制 I 类错误和较高的经验功效，尤其是对于大样本量。我们进一步说明了我们对阿拉伯半岛双变量风数据的测试。