This article gives a synopsis on new developments in affine invariant tests for multivariate normality in an i.i.d.-setting, with special emphasis on asymptotic properties of several classes of weighted \(L^2\)-statistics. Since weighted \(L^2\)-statistics typically have limit normal distributions under fixed alternatives to normality, they open ground for a neighborhood of model validation for normality. The paper also reviews several other invariant tests for this problem, notably the energy test, and it presents the results of a large-scale simulation study. All tests under study are implemented in the accompanying R-package mnt.

中文翻译：

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多元正态性检验-着重于加权$$ L ^ 2 $$ L 2-统计的重要评论

本文概述了仿射不变检验在iid设置中的多元正态性的新发展，特别强调了几类加权\（L ^ 2 \）-统计量的渐近性质。由于加权\（L ^ 2 \）-统计量通常在固定的正态性选择下具有有限的正态分布，因此它们为正态性模型验证的邻居打开了基础。本文还回顾了针对该问题的其他几个不变检验，尤其是能量检验，并给出了大规模仿真研究的结果。所有正在研究的测试都在随附的R- package mnt中进行。