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

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Abstract

LetX 1,X 2, …,X n be independent identically distributed random vectors in IRd,d ⩾ 1, with sample mean\(\bar X_n \) and sample covariance matrixS n. We present a practicable and consistent test for the composite hypothesisH d: the law ofX 1 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 residualsS −1/2 n (X j\(\bar X_n \)) and its pointwise limit exp (−1/2|t|2) underH d. 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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Baringhaus, L., Henze, N. A consistent test for multivariate normality based on the empirical characteristic function. Metrika 35, 339–348 (1988). https://doi.org/10.1007/BF02613322

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