A consistent test for multivariate normality based on the empirical characteristic function
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LetX1,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.
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