We develop a test of normality for spatially indexed functions. The assumption of normality is common in spatial statistics, yet no significance tests, or other means of assessment, have been available for functional data. This paper aims at filling this gap in the case of functional observations on a spatial grid. Our test compares the moments of the spatial (frequency domain) principal component scores to those of a suitable Gaussian distribution. Critical values can be readily obtained from a chi-squared distribution. We provide rigorous theoretical justification for a broad class of weakly stationary functional random fields. We perform simulation studies to assess the the power of the test against various alternatives. An application to Surface Incoming Shortwave Radiation illustrates the practical value of this procedure.

翻译：我们开发了空间索引函数的正常性测试。正常性假设在空间统计中很常见,但对于功能性数据却没有意义测试或其他评估手段。本文件旨在填补空间网格功能观测方面的空白。我们的测试将空间(频域)主要组成部分分数与合适的高斯分布之间的时间比较。关键值可以从一个奇异分布中轻易获得。我们为一大批低静态功能随机字段提供了严格的理论依据。我们进行了模拟研究,以评估该测试相对于各种替代品的威力。对地表侵入短波辐射的应用说明了这一程序的实际价值。