Testing multinormality based on low-dimensional projection

A method based on properties of left-spherical matrix distributions and affine invariant statistics is employed to construct projection tests for multivariate normality. The projection tests are indirectly dependent on the dimension of raw data. As a result, the projection tests can be performed for arbitrary dimension d and sample size n even if n<d in high-dimensional case as soon as the projection dimension is suitably chosen. By Monte Carlo simulation, we show that the projection tests significantly improve the power of existing tests for multinormality in the case of high dimension with a small sample size. Analysis on a practical example shows that the projection tests are useful complements to existing tests for multinormality.