Annals of Statistics

Testing for Normality in Arbitrary Dimension

Sandor Csorgo

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The univariate weak convergence theorem of Murota and Takeuchi (1981) is extended for the Mahalanobis transform of the $d$-variate empirical characteristic function, $d \geq 1$. Then a maximal deviation statistic is proposed for testing the composite hypothesis of $d$-variate normality. Fernique's inequality is used in conjunction with a combination of analytic, numerical analytic, and computer techniques to derive exact upper bounds for the asymptotic percentage points of the statistic. The resulting conservative large sample test is shown to be consistent against every alternative with components having a finite variance. (If $d = 1$ it is consistent against every alternative.) Monte Carlo experiments and the performance of the test on some well-known data sets are also discussed.

Article information

Ann. Statist., Volume 14, Number 2 (1986), 708-723.

First available in Project Euclid: 12 April 2007

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


Primary: 62H15: Hypothesis testing
Secondary: 62F03: Hypothesis testing 62F05: Asymptotic properties of tests

Empirical characteristic function Mahalanobis transform univariate and multivariate normality weak convergence maximal deviation Fernique's and Borell's bounds on the absolute supremum of a Gaussian process


Csorgo, Sandor. Testing for Normality in Arbitrary Dimension. Ann. Statist. 14 (1986), no. 2, 708--723. doi:10.1214/aos/1176349948.

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