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Polar angle tangent vectors follow Cauchy distributions under spherical symmetry

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  • Cacoullos, T.

Abstract

Let X=(X1,…,Xn)′ follow a spherically or elliptically symmetric distribution centered at zero, and Yi=Xi+1/X1, Y=(Y1,…,Yn−1)′. It is shown that under spherical symmetry Y has a symmetric Cauchy distribution and under elliptical symmetry a general Cauchy distribution. Geometrically, Y is the tangent (or cotangent) vector of the polar angle θ1. The simple case of one ratio is treated in Arnold and Brockett (1992), Jones (1999, 2008). Moreover, it is shown that n−1cotθ1 follows the tn−1 distribution, so that the normal theory distributions of Student’s t and correlation coefficient r hold under spherical symmetry.

Suggested Citation

  • Cacoullos, T., 2014. "Polar angle tangent vectors follow Cauchy distributions under spherical symmetry," Journal of Multivariate Analysis, Elsevier, vol. 128(C), pages 147-153.
    • Handle: RePEc:eee:jmvana:v:128:y:2014:i:c:p:147-153
    DOI: 10.1016/j.jmva.2014.03.010
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    References listed on IDEAS

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    1. Cambanis, Stamatis & Huang, Steel & Simons, Gordon, 1981. "On the theory of elliptically contoured distributions," Journal of Multivariate Analysis, Elsevier, vol. 11(3), pages 368-385, September.
    2. Jones, M.C., 2008. "The distribution of the ratio X/Y for all centred elliptically symmetric distributions," Journal of Multivariate Analysis, Elsevier, vol. 99(3), pages 572-573, March.
    3. Cacoullos, Theophilos, 2001. "The F-test of homoscedasticity for correlated normal variables," Statistics & Probability Letters, Elsevier, vol. 54(1), pages 1-3, August.
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    Cited by:

    1. Francq, Christian & Jiménez Gamero, Maria Dolores & Meintanis, Simos, 2015. "Tests for sphericity in multivariate garch models," MPRA Paper 67411, University Library of Munich, Germany.
    2. Albisetti, Isaia & Balabdaoui, Fadoua & Holzmann, Hajo, 2020. "Testing for spherical and elliptical symmetry," Journal of Multivariate Analysis, Elsevier, vol. 180(C).
    3. Francq, C. & Jiménez-Gamero, M.D. & Meintanis, S.G., 2017. "Tests for conditional ellipticity in multivariate GARCH models," Journal of Econometrics, Elsevier, vol. 196(2), pages 305-319.

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