Annals of Statistics
- Ann. Statist.
- Volume 14, Number 2 (1986), 708-723.
Testing for Normality in Arbitrary Dimension
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.
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
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. https://projecteuclid.org/euclid.aos/1176349948