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A new test for multivariate normality


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  • Szekely, Gábor J.
  • Rizzo, Maria L.
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    We propose a new class of rotation invariant and consistent goodness-of-fit tests for multivariate distributions based on Euclidean distance between sample elements. The proposed test applies to any multivariate distribution with finite second moments. In this article we apply the new method for testing multivariate normality when parameters are estimated. The resulting test is affine invariant and consistent against all fixed alternatives. A comparative Monte Carlo study suggests that our test is a powerful competitor to existing tests, and is very sensitive against heavy tailed alternatives.

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    Article provided by Elsevier in its journal Journal of Multivariate Analysis.

    Volume (Year): 93 (2005)
    Issue (Month): 1 (March)
    Pages: 58-80

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    Handle: RePEc:eee:jmvana:v:93:y:2005:i:1:p:58-80

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    Keywords: Goodness-of-fit Strictly negative definite BHEP test Henze-Zirkler test Multivariate skewness Multivariate kurtosis Projection pursuit;


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    1. Romeu, J. L. & Ozturk, A., 1993. "A Comparative Study of Goodness-of-Fit Tests for Multivariate Normality," Journal of Multivariate Analysis, Elsevier, vol. 46(2), pages 309-334, August.
    2. Thas, O. & Ottoy, J. P., 2003. "Some generalizations of the Anderson-Darling statistic," Statistics & Probability Letters, Elsevier, vol. 64(3), pages 255-261, September.
    3. L. Baringhaus & N. Henze, 1988. "A consistent test for multivariate normality based on the empirical characteristic function," Metrika, Springer, vol. 35(1), pages 339-348, December.
    4. Henze, Norbert & Wagner, Thorsten, 1997. "A New Approach to the BHEP Tests for Multivariate Normality," Journal of Multivariate Analysis, Elsevier, vol. 62(1), pages 1-23, July.
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    Cited by:
    1. Diks, C.G.H. & Panchenko, V., 2005. "Nonparametric Tests for Serial Independence Based on Quadratic Forms," CeNDEF Working Papers 05-13, Universiteit van Amsterdam, Center for Nonlinear Dynamics in Economics and Finance.
    2. Cuesta-Albertos, J.A. & del Barrio, E. & Fraiman, R. & Matran, C., 2007. "The random projection method in goodness of fit for functional data," Computational Statistics & Data Analysis, Elsevier, vol. 51(10), pages 4814-4831, June.
    3. Huang, Yufen & Kao, Tzu-Ling & Wang, Tai-Ho, 2007. "Influence functions and local influence in linear discriminant analysis," Computational Statistics & Data Analysis, Elsevier, vol. 51(8), pages 3844-3861, May.
    4. Panagiotelis, Anastasios & Smith, Michael, 2008. "Bayesian density forecasting of intraday electricity prices using multivariate skew t distributions," International Journal of Forecasting, Elsevier, vol. 24(4), pages 710-727.
    5. Barigozzi, Matteo & Alessi, Lucia & Capasso, Marco & Fagiolo, Giorgio, 2009. "The distribution of households consumption-expenditure budget shares," Working Paper Series 1061, European Central Bank.
    6. Tenreiro, Carlos, 2011. "An affine invariant multiple test procedure for assessing multivariate normality," Computational Statistics & Data Analysis, Elsevier, vol. 55(5), pages 1980-1992, May.
    7. Araújo, Tanya & Dias, João & Eleutério, Samuel & Louçã, Francisco, 2013. "A measure of multivariate kurtosis for the identification of the dynamics of a N-dimensional market," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 392(17), pages 3708-3714.
    8. Cees Diks & Valentyn Panchenko, 2005. "Nonparametric Tests for Serial Independence Based on Quadratic Forms," Tinbergen Institute Discussion Papers 05-076/1, Tinbergen Institute.
    9. Jörg-Peter Schräpler, 2011. "Konstruktion von SGB II – Dichten als Raumindikator und ihre Verwendung als Indikator im Rahmen der Sozialberichterstattung am Beispiel der „sozialen Belastung“ von Schulstandorten in NRW – ei," AStA Wirtschafts- und Sozialstatistisches Archiv, Springer, vol. 5(2), pages 97-124, August.


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