A consistent test for multivariate normality based on the empirical characteristic function

AbstractLetX1,X2, …,Xn be independent identically distributed random vectors in IRd,d ⩾ 1, with sample mean $$\bar X_n $$ and sample covariance matrixSn. We present a practicable and consistent test for the composite hypothesisHd: the law ofX1 is a non-degenerate normal distribution, based on a weighted integral of the squared modulus of the difference between the empirical characteristic function of the residualsSn−1/2(Xj − $$\bar X_n $$ ) and its pointwise limit exp (−1/2|t|2) underHd. The limiting null distribution of the test statistic is obtained, and a table with critical values for various choices ofn andd based on extensive simulations is supplied.