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A new test for multivariate normality


  • Szekely, Gábor J.
  • Rizzo, Maria L.


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.

Suggested Citation

  • Szekely, Gábor J. & Rizzo, Maria L., 2005. "A new test for multivariate normality," Journal of Multivariate Analysis, Elsevier, vol. 93(1), pages 58-80, March.
  • Handle: RePEc:eee:jmvana:v:93:y:2005:i:1:p:58-80

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    References listed on IDEAS

    1. Thas, O. & Ottoy, J. P., 2003. "Some generalizations of the Anderson-Darling statistic," Statistics & Probability Letters, Elsevier, vol. 64(3), pages 255-261, September.
    2. L. Baringhaus & N. Henze, 1988. "A consistent test for multivariate normality based on the empirical characteristic function," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 35(1), pages 339-348, December.
    3. 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.
    4. 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.
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    Cited by:

    1. Jun Li & Yao Yu, 2015. "A Nonparametric Test of Missing Completely at Random for Incomplete Multivariate Data," Psychometrika, Springer;The Psychometric Society, vol. 80(3), pages 707-726, September.
    2. Ebner, Bruno, 2012. "Asymptotic theory for the test for multivariate normality by Cox and Small," Journal of Multivariate Analysis, Elsevier, vol. 111(C), pages 368-379.
    3. Barigozzi, Matteo & Alessi, Lucia & Capasso, Marco & Fagiolo, Giorgio, 2012. "The distribution of household consumption-expenditure budget shares," Structural Change and Economic Dynamics, Elsevier, vol. 23(1), pages 69-91.
    4. 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 – ein Kerndi," AStA Wirtschafts- und Sozialstatistisches Archiv, Springer;Deutsche Statistische Gesellschaft - German Statistical Society, vol. 5(2), pages 97-124, August.
    5. Sandeep R. Chandukala & Jeffrey P. Dotson & Jeff D. Brazell & Greg M. Allenby, 2011. "Bayesian Analysis of Hierarchical Effects," Marketing Science, INFORMS, vol. 30(1), pages 123-133, 01-02.
    6. 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.
    7. Ravi Kashyap, 2016. "Combining Dimension Reduction, Distance Measures and Covariance," Papers 1603.09060,, revised Nov 2017.
    8. Meintanis, Simos G. & Ngatchou-Wandji, Joseph & Taufer, Emanuele, 2015. "Goodness-of-fit tests for multivariate stable distributions based on the empirical characteristic function," Journal of Multivariate Analysis, Elsevier, vol. 140(C), pages 171-192.
    9. 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.
    10. Cees Diks & Valentyn Panchenko, 2005. "Nonparametric Tests for Serial Independence Based on Quadratic Forms," Tinbergen Institute Discussion Papers 05-076/1, Tinbergen Institute.
    11. 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.
    12. Simos G. Meintanis & James Allison & Leonard Santana, 2016. "Goodness-of-fit tests for semiparametric and parametric hypotheses based on the probability weighted empirical characteristic function," Statistical Papers, Springer, vol. 57(4), pages 957-976, December.
    13. Patra, Rohit K. & Sen, Bodhisattva & Székely, Gábor J., 2016. "On a nonparametric notion of residual and its applications," Statistics & Probability Letters, Elsevier, vol. 109(C), pages 208-213.
    14. Rizzo, Maria L. & Haman, John T., 2016. "Expected distances and goodness-of-fit for the asymmetric Laplace distribution," Statistics & Probability Letters, Elsevier, vol. 117(C), pages 158-164.
    15. 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.
    16. 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.
    17. Dutta, Subhajit & Genton, Marc G., 2014. "A non-Gaussian multivariate distribution with all lower-dimensional Gaussians and related families," Journal of Multivariate Analysis, Elsevier, vol. 132(C), pages 82-93.


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