A class of invariant consistent tests for multivariate normality

Let be independent identically distributed random vectors in Rd d ≥ 1 , with sample mean [Xbar] n and sample covariance matrix S n . We present a class of practicable afflne-invariant tests for the composite hypothesis H d the law of X 1 is a non-degenerate normal distribution which are consistent against any fixed non- normal alternative distribution. The test statistic is a weighted integral of the squared modulus of the difference between the empirical characteristic function of the scaled residuals and its pointwise limit under H d - An alternative representation is given in terms of an L 2-distance between densities. The limiting null distribution of the test statistic is obtained. Power performance of the new tests is assessed in a Monte Carlo study.