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Affine Invariant Multivariate Sign and Rank Tests and Corresponding Estimates: a Review

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  • Hannu Oja

Abstract

The paper reviews recent contributions to the statistical inference methods, tests and estimates, based on the generalized median of Oja. Multivariate analogues of sign and rank concepts, affine invariant one‐sample and two‐sample sign tests and rank tests, affine equivariant median and Hodges–Lehmann‐type estimates are reviewed and discussed. Some comparisons are made to other generalizations. The theory is illustrated by two examples.

Suggested Citation

  • Hannu Oja, 1999. "Affine Invariant Multivariate Sign and Rank Tests and Corresponding Estimates: a Review," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 26(3), pages 319-343, September.
    • Handle: RePEc:bla:scjsta:v:26:y:1999:i:3:p:319-343
    DOI: 10.1111/1467-9469.00152
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    Cited by:

    1. 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.
    2. Wellmann, Robin & Müller, Christine H., 2010. "Tests for multiple regression based on simplicial depth," Journal of Multivariate Analysis, Elsevier, vol. 101(4), pages 824-838, April.
    3. Karl Mosler, 2003. "Central Regions and Dependency," Methodology and Computing in Applied Probability, Springer, vol. 5(1), pages 5-21, March.
    4. repec:jss:jstsof:43:i05 is not listed on IDEAS
    5. Harrar, Solomon W. & Bathke, Arne C., 2008. "Nonparametric methods for unbalanced multivariate data and many factor levels," Journal of Multivariate Analysis, Elsevier, vol. 99(8), pages 1635-1664, September.
    6. Barone, P., 2016. "Bivariate one-sample optimal location test for spherical stable densities by Pade’ methods," Journal of Multivariate Analysis, Elsevier, vol. 144(C), pages 189-199.
    7. Eustasio Del Barrio & Alberto Gonzalez-Sanz & Marc Hallin, 2019. "A Note on the Regularity of Center-Outward Distribution and Quantile Functions," Working Papers ECARES 2019-33, ULB -- Universite Libre de Bruxelles.
    8. Nadar, M. & Hettmansperger, T. P. & Oja, H., 2003. "The asymptotic covariance matrix of the Oja median," Statistics & Probability Letters, Elsevier, vol. 64(4), pages 431-442, October.
    9. Biau, Gérard & Devroye, Luc & Dujmović, Vida & Krzyżak, Adam, 2012. "An affine invariant k-nearest neighbor regression estimate," Journal of Multivariate Analysis, Elsevier, vol. 112(C), pages 24-34.
    10. Möttönen, J. & Hüsler, J. & Oja, H., 2003. "Multivariate nonparametric tests in a randomized complete block design," Journal of Multivariate Analysis, Elsevier, vol. 85(1), pages 106-129, April.
    11. Jin Wang & Weihua Zhou, 2015. "Effect of kurtosis on efficiency of some multivariate medians," Journal of Nonparametric Statistics, Taylor & Francis Journals, vol. 27(3), pages 331-348, September.
    12. Hudecová, Šárka & Šiman, Miroslav, 2022. "Multivariate ranks based on randomized lift-interdirections," Computational Statistics & Data Analysis, Elsevier, vol. 172(C).
    13. Ollila, Esa & Oja, Hannu & Croux, Christophe, 2003. "The affine equivariant sign covariance matrix: asymptotic behavior and efficiencies," Journal of Multivariate Analysis, Elsevier, vol. 87(2), pages 328-355, November.
    14. del Barrio, Eustasio & González-Sanz, Alberto & Hallin, Marc, 2020. "A note on the regularity of optimal-transport-based center-outward distribution and quantile functions," Journal of Multivariate Analysis, Elsevier, vol. 180(C).
    15. Sakineh Dehghan & Mohammad Reza Faridrohani, 2019. "Affine invariant depth-based tests for the multivariate one-sample location problem," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 28(3), pages 671-693, September.

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