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Testing for elliptical symmetry in covariance-matrix-based analyses

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  • Schott, James R.

Abstract

Many normal-theory test procedures for covariance matrices remain valid outside the family of normal distributions if the matrix of fourth-order moments has structure similar to that of a normal distribution. In particular, for elliptical distributions this matrix of fourth-order moments is a scalar multiple of that for the normal, and for this reason many normal-theory statistics can be adjusted by a scalar multiple so as to retain their asymptotic distributional properties across elliptical distributions. For these analyses, a test for the validity of these scalar-adjusted normal-theory procedures can be viewed as a test on the structure of the matrix of fourth-order moments. In this paper, we develop a Wald statistic for conducting such a test.

Suggested Citation

  • Schott, James R., 2002. "Testing for elliptical symmetry in covariance-matrix-based analyses," Statistics & Probability Letters, Elsevier, vol. 60(4), pages 395-404, December.
    • Handle: RePEc:eee:stapro:v:60:y:2002:i:4:p:395-404
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    References listed on IDEAS

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    1. J. P. Royston, 1983. "Some Techniques for Assessing Multivarate Normality Based on the Shapiro‐Wilk W," Journal of the Royal Statistical Society Series C, Royal Statistical Society, vol. 32(2), pages 121-133, June.
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    1. Ali Genç, 2013. "Moments of truncated normal/independent distributions," Statistical Papers, Springer, vol. 54(3), pages 741-764, August.
    2. Sakineh Dehghan & Mohammad Reza Faridrohani & Zahra Barzegar, 2023. "Testing for diagonal symmetry based on center-outward ranking," Statistical Papers, Springer, vol. 64(1), pages 255-283, February.
    3. Norbert Henze & Celeste Mayer, 2020. "More good news on the HKM test for multivariate reflected symmetry about an unknown centre," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 72(3), pages 741-770, June.
    4. Niu, Lu & Liu, Xiumin & Zhao, Junlong, 2020. "Robust estimator of the correlation matrix with sparse Kronecker structure for a high-dimensional matrix-variate," Journal of Multivariate Analysis, Elsevier, vol. 177(C).
    5. Jiajuan Liang & Kai Wang Ng & Guoliang Tian, 2019. "A class of uniform tests for goodness-of-fit of the multivariate $$L_p$$ L p -norm spherical distributions and the $$l_p$$ l p -norm symmetric distributions," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 71(1), pages 137-162, February.
    6. Sladana Babic & Laetitia Gelbgras & Marc Hallin & Christophe Ley, 2019. "Optimal tests for elliptical symmetry: specified and unspecified location," Working Papers ECARES 2019-26, ULB -- Universite Libre de Bruxelles.
    7. Frank Schuhmacher & Hendrik Kohrs & Benjamin R. Auer, 2021. "Justifying Mean-Variance Portfolio Selection when Asset Returns Are Skewed," Management Science, INFORMS, vol. 67(12), pages 7812-7824, December.
    8. Batsidis, Apostolos & Zografos, Konstantinos, 2013. "A necessary test of fit of specific elliptical distributions based on an estimator of Song’s measure," Journal of Multivariate Analysis, Elsevier, vol. 113(C), pages 91-105.
    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. Schott, James R., 2003. "Kronecker product permutation matrices and their application to moment matrices of the normal distribution," Journal of Multivariate Analysis, Elsevier, vol. 87(1), pages 177-190, October.
    11. Boente, Graciela & Salibián Barrera, Matías & Tyler, David E., 2014. "A characterization of elliptical distributions and some optimality properties of principal components for functional data," Journal of Multivariate Analysis, Elsevier, vol. 131(C), pages 254-264.

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