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We use a system of first-order partial differential equations thatcharacterize the moment generating function of the $d$-variate standard normaldistribution to construct a class of affine invariant tests for normality inany dimension. We derive the limit null distribution of the resulting teststatistics, and we prove consistency of the tests against general alternatives.In the case $d > 1$, a certain limit of these tests is connected with twomeasures of multivariate skewness. The new tests show strong power performancewhen compared to well-known competitors, especially against heavy-taileddistributions, and they are illustrated by means of a real data set.

Testing for normality in any dimension based on a partial differential equation involving the moment generating function