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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 way 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 also outline topics for further research.

The recently introduced framework of universal inference provides a new approach to constructing hypothesis tests and confidence regions that are valid in finite samples and do not rely on any specific regularity assumptions on the underlying statistical model. At the core of the methodology is a split likelihood ratio statistic, which is formed under data splitting and compared to a cleverly selected universal critical value. As this critical value can be very conservative, it is interesting to mitigate the potential loss of power by careful choice of the ratio according to which data are split. Motivated by this problem, we study the split likelihood ratio test under local alternatives and introduce the resulting class of noncentral split chi-square distributions. We investigate the properties of this new class of distributions and use it to numerically examine and propose an optimal choice of the data splitting ratio for tests of composite hypotheses of different dimensions.

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