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Tests and estimates of shape based on spatial signs and ranks

Author

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  • Seija Sirkiä
  • Sara Taskinen
  • Hannu Oja
  • David Tyler

Abstract

Nonparametric procedures for testing and estimation of the shape matrix in the case of multivariate elliptic distribution are considered. Testing for sphericity is an important special case. The tests and estimates are based on the spatial sign and rank covariance matrices. The estimates based on the spatial sign covariance matrix and symmetrized spatial sign covariance matrix are Tyler's [A distribution-free M-estimator of multivariate scatter, Ann. Statist. 15 (1987), pp. 234–251] shape matrix and and Dümbgen's [On Tyler's M-functional of scatter in high dimension, Ann. Inst. Statist. Math. 50 (1998), pp. 471–491] shape matrix, respectively. The test based on the spatial sign covariance matrix is the sign test statistic in the class of nonparametric tests proposed by Hallin and Paindaveine [Semiparametrically efficient rank-based inference for shape. I. Optimal rank-based tests for sphericity, Ann. Statist. 34 (2006), pp. 2707–2756]. New tests and estimates based on the spatial rank covariance matrix are proposed. The shape estimates introduced in the paper play an important role in the inner standardisation of the spatial sign and rank tests for multivariate location. Limiting distributions of the tests and estimates are reviewed and derived, and asymptotic efficiencies as well as finite-sample efficiencies of the proposed tests are compared with those of the classical modified John's [Some optimal multivariate tests, Biometrika 58 (1971), pp. 123–127; The distribution of a statistic used for testing sphericity of normal distributions, Biometrika 59 (1972), pp. 169–173] test and the van der Waerden test (Hallin and Paindaveine, [Semiparametrically efficient rank-based inference for shape. I. Optimal rank-based tests for sphericity, Ann. Statist. 34 (2006), pp. 2707–2756]). The symmetrised spatial sign- and rank-based estimates and tests seem to have a very high efficiency in the multivariate normal case, and they are much better than the classical estimate (shape matrix based on the regular covariance matrix) and test (John's test) for distributions with heavy tails.

Suggested Citation

  • Seija Sirkiä & Sara Taskinen & Hannu Oja & David Tyler, 2009. "Tests and estimates of shape based on spatial signs and ranks," Journal of Nonparametric Statistics, Taylor & Francis Journals, vol. 21(2), pages 155-176.
    • Handle: RePEc:taf:gnstxx:v:21:y:2009:i:2:p:155-176
    DOI: 10.1080/10485250802495691
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    References listed on IDEAS

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    1. Thomas P. Hettmansperger, 2002. "A practical affine equivariant multivariate median," Biometrika, Biometrika Trust, vol. 89(4), pages 851-860, December.
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    4. Lutz Dümbgen & David E. Tyler, 2005. "On the Breakdown Properties of Some Multivariate M‐Functionals," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 32(2), pages 247-264, June.
    5. Marden, John I., 1999. "Some robust estimates of principal components," Statistics & Probability Letters, Elsevier, vol. 43(4), pages 349-359, July.
    6. Paindaveine, Davy, 2008. "A canonical definition of shape," Statistics & Probability Letters, Elsevier, vol. 78(14), pages 2240-2247, October.
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    Cited by:

    1. Raymaekers, Jakob & Rousseeuw, Peter, 2019. "A generalized spatial sign covariance matrix," Journal of Multivariate Analysis, Elsevier, vol. 171(C), pages 94-111.
    2. Dürre, Alexander & Vogel, Daniel & Fried, Roland, 2015. "Spatial sign correlation," Journal of Multivariate Analysis, Elsevier, vol. 135(C), pages 89-105.
    3. Davy Paindaveine & Germain Van Bever, 2017. "Halfspace Depths for Scatter, Concentration and Shape Matrices," Working Papers ECARES ECARES 2017-19, ULB -- Universite Libre de Bruxelles.
    4. C. Croux & C. Dehon & A. Yadine, 2010. "The k-step spatial sign covariance matrix," Advances in Data Analysis and Classification, Springer;German Classification Society - Gesellschaft für Klassifikation (GfKl);Japanese Classification Society (JCS);Classification and Data Analysis Group of the Italian Statistical Society (CLADAG);International Federation of Classification Societies (IFCS), vol. 4(2), pages 137-150, September.
    5. Dürre, Alexander & Vogel, Daniel & Tyler, David E., 2014. "The spatial sign covariance matrix with unknown location," Journal of Multivariate Analysis, Elsevier, vol. 130(C), pages 107-117.

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