We study a novel class of affine invariant and consistent tests for normality in any dimension. The tests are based on a characterization of the standard $d$-variate normal distribution as the unique solution of an initial value problem of a partial differential equation motivated by the harmonic oscillator, which is a special case of a Schrodinger operator. We derive the asymptotic distribution of the test statistics under the hypothesis of normality as well as under fixed and contiguous alternatives. The tests are consistent against general alternatives, exhibit strong power performance for finite samples, and they are applied to a classical data set due to R.A. Fisher. The results can also be used for a neighborhood-of-model validation procedure.

中文翻译：

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通过特征函数空间中谐振子的零点测试多元正态性

我们研究了一类新的仿射不变性和任何维度的正态性一致性检验。测试基于标准 $d$-variate 正态分布的特征，作为由谐振子激励的偏微分方程的初值问题的唯一解，这是 Schrodinger 算子的一个特例。我们在正态性假设以及固定和连续替代方案下推导出检验统计量的渐近分布。这些测试与一般替代方案一致，对有限样本表现出强大的功效，并且由于 RA Fisher，它们被应用于经典数据集。结果也可用于模型邻域验证程序。