  We study a novel class of affine invariant and consistent tests for
multivariate normality. The tests are based on a characterization of the
standard $d$-variate normal distribution by means of the unique solution of an
initial value problem connected to a partial differential equation, which is
motivated by a multivariate Stein equation. The test criterion is a suitably
weighted $L^2$-statistic. We derive the limit distribution of the test
statistic under the null hypothesis as well as under contiguous and fixed
alternatives to normality. A consistent estimator of the limiting variance
under fixed alternatives as well as an asymptotic confidence interval of the
distance of an underlying alternative with respect to the multivariate normal
law is derived. In simulation studies, we show that the tests are strong in
comparison with prominent competitors, and that the empirical coverage rate of
the asymptotic confidence interval converges to the nominal level. We present a
real data example, and we outline topics for further research.
