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Optimal tests for elliptical symmetry: specified and unspecified location

Author

Listed:
  • Sladana Babic
  • Laetitia Gelbgras
  • Marc Hallin
  • Christophe Ley

Abstract

Although the assumption of elliptical symmetry is quite common in multivariate analysis and widespread in a number of applications, the problem of testing the null hypothesis of ellipticity so far has not been addressed in a fully satisfactory way. Most of the literature in the area indeed addresses the null hypothesis of elliptical symmetry with specified location and actually addresses location rather than non-elliptical alternatives. In thi spaper, we are proposing new classes of testing procedures,both for specified and unspecified location. The backbone of our construction is Le Cam’s asymptotic theory of statistical experiments, and optimality is to be understood locally and asymptotically within the family of generalized skew-elliptical distributions. The tests we are proposing are meeting all the desired properties of a “good” test of elliptical symmetry:they have a simple asymptotic distribution under the entire null hypothesis of elliptical symmetry with unspecified radial density and shape parameter; they are affine-invariant, computationally fast, intuitively understandable, and not too demanding in terms of moments. While achieving optimality against generalized skew-elliptical alternatives, they remain quite powerful under a much broader class of non-elliptical distributions and significantly outperform the available competitors

Suggested Citation

  • 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.
  • Handle: RePEc:eca:wpaper:2013/295909
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    References listed on IDEAS

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    1. Yves Dominicy & Pauliina Ilmonen & David Veredas, 2017. "Multivariate Hill Estimators," International Statistical Review, International Statistical Institute, vol. 85(1), pages 108-142, April.
    2. Swensen, Anders Rygh, 1985. "The asymptotic distribution of the likelihood ratio for autoregressive time series with a regression trend," Journal of Multivariate Analysis, Elsevier, vol. 16(1), pages 54-70, February.
    3. repec:ulb:ulbeco:2013/127941 is not listed on IDEAS
    4. D. Vogel & R. Fried, 2011. "Elliptical graphical modelling," Biometrika, Biometrika Trust, vol. 98(4), pages 935-951.
    5. Hallin, Marc & Paindaveine, Davy, 2005. "Affine-invariant aligned rank tests for the multivariate general linear model with VARMA errors," Journal of Multivariate Analysis, Elsevier, vol. 93(1), pages 122-163, March.
    6. Bernard Garel & Marc Hallin, 1995. "Local asymptotic normality of multivariate ARMA processes with a linear trend," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 47(3), pages 551-579, September.
    7. Liebscher, Eckhard, 2005. "A semiparametric density estimator based on elliptical distributions," Journal of Multivariate Analysis, Elsevier, vol. 92(1), pages 205-225, January.
    8. Huffer, Fred W. & Park, Cheolyong, 2007. "A test for elliptical symmetry," Journal of Multivariate Analysis, Elsevier, vol. 98(2), pages 256-281, February.
    9. Douglas J. Hodgson & Oliver Linton & Keith Vorkink, 2002. "Testing the capital asset pricing model efficiently under elliptical symmetry: a semiparametric approach," Journal of Applied Econometrics, John Wiley & Sons, Ltd., vol. 17(6), pages 617-639, December.
    10. Schott, James R., 2002. "Testing for elliptical symmetry in covariance-matrix-based analyses," Statistics & Probability Letters, Elsevier, vol. 60(4), pages 395-404, December.
    11. M. C. Jones & Arthur Pewsey, 2009. "Sinh-arcsinh distributions," Biometrika, Biometrika Trust, vol. 96(4), pages 761-780.
    12. Lombardi, Marco J. & Veredas, David, 2009. "Indirect estimation of elliptical stable distributions," Computational Statistics & Data Analysis, Elsevier, vol. 53(6), pages 2309-2324, April.
    13. Marc Hallin & Davy Paindaveine & Thomas Verdebout, 2010. "Testing for Common Principal Components under Heterokurticity," Journal of Nonparametric Statistics, Taylor & Francis Journals, vol. 22(7), pages 879-895.
    14. Marc Hallin & Davy Paindaveine & Thomas Verdebout, 2009. "Optimal rank-based testing for principal component," Working Papers ECARES 2009_013, ULB -- Universite Libre de Bruxelles.
    15. repec:eca:wpaper:2013/128686 is not listed on IDEAS
    16. Keith Vorkink & Douglas J. Hodgson & Oliver Linton, 2002. "Testing the capital asset pricing model efficiently under elliptical symmetry: a semiparametric approach," Journal of Applied Econometrics, John Wiley & Sons, Ltd., vol. 17(6), pages 617-639.
    17. Marc Genton & Nicola Loperfido, 2005. "Generalized skew-elliptical distributions and their quadratic forms," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 57(2), pages 389-401, June.
    18. Ley, Christophe & Paindaveine, Davy, 2010. "On the singularity of multivariate skew-symmetric models," Journal of Multivariate Analysis, Elsevier, vol. 101(6), pages 1434-1444, July.
    19. Wraith, Darren & Forbes, Florence, 2015. "Location and scale mixtures of Gaussians with flexible tail behaviour: Properties, inference and application to multivariate clustering," Computational Statistics & Data Analysis, Elsevier, vol. 90(C), pages 61-73.
    20. Rainer Dyckerhoff & Christophe Ley & Davy Paindaveine, 2014. "Depth-Based Runs Tests for bivariate Central Symmetry," Working Papers ECARES ECARES 2014-03, ULB -- Universite Libre de Bruxelles.
    21. A. Azzalini & A. Capitanio, 1999. "Statistical applications of the multivariate skew normal distribution," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 61(3), pages 579-602.
    22. Branco, Márcia D. & Dey, Dipak K., 2001. "A General Class of Multivariate Skew-Elliptical Distributions," Journal of Multivariate Analysis, Elsevier, vol. 79(1), pages 99-113, October.
    23. Adelchi Azzalini & Antonella Capitanio, 2003. "Distributions generated by perturbation of symmetry with emphasis on a multivariate skew t‐distribution," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 65(2), pages 367-389, May.
    24. Manzotti, A. & Pérez, Francisco J. & Quiroz, Adolfo J., 2002. "A Statistic for Testing the Null Hypothesis of Elliptical Symmetry," Journal of Multivariate Analysis, Elsevier, vol. 81(2), pages 274-285, May.
    25. Marc Hallin & Davy Paindaveine & Thomas Verdebout, 2011. "Optimal Rank-Based Tests for Common Principal Components," Working Papers ECARES ECARES 2011-032, ULB -- Universite Libre de Bruxelles.
    26. Hallin, Marc & Paindaveine, Davy, 2009. "Optimal tests for homogeneity of covariance, scale, and shape," Journal of Multivariate Analysis, Elsevier, vol. 100(3), pages 422-444, March.
    27. Sakhanenko, Lyudmila, 2008. "Testing for ellipsoidal symmetry: A comparison study," Computational Statistics & Data Analysis, Elsevier, vol. 53(2), pages 565-581, December.
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    Keywords

    Elliptical Symmetry; Local Asymptotic normality; Maximin tests; Multivariate skewness; semiparametric inference; skew-elliptical densities;

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