A class of invariant consistent tests for multivariate normality

Let X$_1$,..., X$_n$ be independent identically distributed random vectors in R$^d$ 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 S$_n$$^{-\frac{1}{2}}$(X$_j$ - X$_n$) and its pointwise limit exp(-$\frac{1}{2}||t||^2)$ 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.