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A New Approach to the BHEP Tests for Multivariate Normality

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  • Henze, Norbert
  • Wagner, Thorsten
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    Abstract

    LetX1, ..., Xnbe i.i.d. randomd-vectors,d[greater-or-equal, slanted]1, with sample meanXand sample covariance matrixS. For testing the hypothesisHdthat the law ofX1is some nondegenerate normal distribution, there is a whole class of practicable affine invariant and universally consistent tests. These procedures are based on weighted integrals of the squared modulus of the difference between the empirical characteristic function of the scaled residualsYj=S-1/2(Xj-X) and its almost sure pointwise limit exp(-||t||2/2) underHd. The test statistics have an alternative interpretation in terms ofL2-distances between a nonparametric kernel density estimator and the parametric density estimator underHd, applied toY1, ..., Yn. By working in the Fréchet space of continuous functions on d, we obtain a new representation of the limiting null distributions of the test statistics and show that the tests have asymptotic power against sequences of contiguous alternatives converging toHdat the raten-1/2, independent ofd.

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    Bibliographic Info

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

    Volume (Year): 62 (1997)
    Issue (Month): 1 (July)
    Pages: 1-23

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    Handle: RePEc:eee:jmvana:v:62:y:1997:i:1:p:1-23

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    Keywords: test for multivariate normality empirical characteristic function Gaussian process contiguous alternatives;

    References

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    1. 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.
    2. Henze, Norbert, 1997. "Extreme smoothing and testing for multivariate normality," Statistics & Probability Letters, Elsevier, vol. 35(3), pages 203-213, October.
    3. 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.
    4. Sándor Csörgő, 1989. "Consistency of some tests for multivariate normality," Metrika, Springer, vol. 36(1), pages 107-116, December.
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    Cited by:
    1. Muneya Matsui & Akimichi Takemura, 2008. "Goodness-of-fit tests for symmetric stable distributions—Empirical characteristic function approach," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer, vol. 17(3), pages 546-566, November.
    2. Tanya Ara\'ujo & Jo\~ao Dias & Samuel Eleut\'erio & Francisco Lou\c{c}\~a, 2012. "How Fama Went Wrong: Measures of Multivariate Kurtosis for the Identification of the Dynamics of a N-Dimensional Market," Papers 1207.1202, arXiv.org.
    3. Bilodeau, M. & Lafaye de Micheaux, P., 2005. "A multivariate empirical characteristic function test of independence with normal marginals," Journal of Multivariate Analysis, Elsevier, vol. 95(2), pages 345-369, August.
    4. Klar, B. & Lindner, F. & Meintanis, S.G., 2012. "Specification tests for the error distribution in GARCH models," Computational Statistics & Data Analysis, Elsevier, vol. 56(11), pages 3587-3598.
    5. Baringhaus, Ludwig & Taherizadeh, Fatemeh, 2010. "Empirical Hankel transforms and its applications to goodness-of-fit tests," Journal of Multivariate Analysis, Elsevier, vol. 101(6), pages 1445-1457, July.
    6. 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.
    7. Henze, N. & Klar, B. & Meintanis, S. G., 2003. "Invariant tests for symmetry about an unspecified point based on the empirical characteristic function," Journal of Multivariate Analysis, Elsevier, vol. 87(2), pages 275-297, November.
    8. Gupta, A. K. & Henze, N. & Klar, B., 2004. "Testing for affine equivalence of elliptically symmetric distributions," Journal of Multivariate Analysis, Elsevier, vol. 88(2), pages 222-242, February.
    9. Huffer, Fred W. & Park, Cheolyong, 2007. "A test for elliptical symmetry," Journal of Multivariate Analysis, Elsevier, vol. 98(2), pages 256-281, February.
    10. Tan, Ming & Fang, Hong-Bin & Tian, Guo-Liang & Wei, Gang, 2005. "Testing multivariate normality in incomplete data of small sample size," Journal of Multivariate Analysis, Elsevier, vol. 93(1), pages 164-179, March.
    11. 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.
    12. Tanya Araujo & João Dias & Samuel Eleutério & Francisco Louçã, 2012. "How Fama Went Wrong: Measures of Multivariate Kurtosis for the Identification of the Dynamics of a N-Dimensional Market," Working Papers Department of Economics 2012/21, ISEG - School of Economics and Management, Department of Economics, University of Lisbon.
    13. Tenreiro, Carlos, 2009. "On the choice of the smoothing parameter for the BHEP goodness-of-fit test," Computational Statistics & Data Analysis, Elsevier, vol. 53(4), pages 1038-1053, February.
    14. Norbert Henze, 2002. "Invariant tests for multivariate normality: a critical review," Statistical Papers, Springer, vol. 43(4), pages 467-506, October.
    15. Epps, T. W., 1999. "Limiting behavior of the ICF test for normality under Gram-Charlier alternatives," Statistics & Probability Letters, Elsevier, vol. 42(2), pages 175-184, April.
    16. Muneya Matsui & Akimichi Takemura, 2005. "Goodness-of-Fit Tests for Symmetric Stable Distributions - Empirical Characteristic Function Approach," CIRJE F-Series CIRJE-F-384, CIRJE, Faculty of Economics, University of Tokyo.
    17. Norbert Henze & Bernhard Klar, 2002. "Goodness-of-Fit Tests for the Inverse Gaussian Distribution Based on the Empirical Laplace Transform," Annals of the Institute of Statistical Mathematics, Springer, vol. 54(2), pages 425-444, June.
    18. Nora Gürtler & Norbert Henze, 2000. "Goodness-of-Fit Tests for the Cauchy Distribution Based on the Empirical Characteristic Function," Annals of the Institute of Statistical Mathematics, Springer, vol. 52(2), pages 267-286, June.
    19. Jiménez-Gamero, M.D. & Alba-Fernández, V. & Muñoz-García, J. & Chalco-Cano, Y., 2009. "Goodness-of-fit tests based on empirical characteristic functions," Computational Statistics & Data Analysis, Elsevier, vol. 53(12), pages 3957-3971, October.
    20. N. Balakrishnan & M. Brito & A. Quiroz, 2013. "On the goodness-of-fit procedure for normality based on the empirical characteristic function for ranked set sampling data," Metrika, Springer, vol. 76(2), pages 161-177, February.

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