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.
Similar content being viewed by others
References
Baringhaus L, Danschke R, Henze N (1988) Recent and classical tests for normality — a comparative study. Comm Statist B Comp Simul (submitted)
Cox DR, Small NJH (1978) Testing multivariate normality. Biometrika 65:263–272
Csörgő S (1986) Testing for normality in arbitrary dimension. Ann Statist 14:708–723
De Wet T, Randles RH (1987) On the effect of substituting parameter estimators in limitingx 2,U andV statistics. Ann Statist 15:398–412
Eaton MR, Perlman MD (1973) The non-singularity of generalized sample covariance matrices. Ann Statist 1:710–717
Epps TW, Pulley LB (1983) A test for normality based on the empirical characteristic function. Biometrika 70:723–726
Fisher RA (1936) The use of multiple measurements in taxonomic problems. Ann Eugenics 7/II:179–188
Gnanadesikan R (1977) Methods for statistical data analysis of multivariate observations. Wiley, New York.
Koziol JA (1982) A class of invariant procedures for assessing multivariate normality. Biometrika 69:423–427
Koziol JA (1983) On assessing multivariate normality. J Roy Statist Soc Ser B 45:358–361
Mardia KV (1980) Tests of univariate and multivariate normality. In: Krishnaiah PR (ed) Handbook of statistics, I Analysis of variance. North-Holland, Amsterdam, pp 279–320
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
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
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF02613322