The Annals of Statistics

Testing for Normality in Arbitrary Dimension

Sandor Csorgo

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Abstract

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

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

Dates
First available in Project Euclid: 12 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aos/1176349948

Digital Object Identifier
doi:10.1214/aos/1176349948

Mathematical Reviews number (MathSciNet)
MR840524

Zentralblatt MATH identifier
0615.62060

JSTOR
links.jstor.org

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

Keywords
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

Citation

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


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