Tests for multivariate normality—a critical review with emphasis on weighted $$L^2$$-statistics

Abstract

AbstractThis 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 $$L^2$$ L 2 -statistics. Since weighted $$L^2$$ 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 -package .