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Equivariance and invariance properties of multivariate quantile and related functions, and the role of standardisation

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  • Robert Serfling

Abstract

Equivariance and invariance issues arise as a fundamental but often problematic aspect of multivariate statistical analysis. For multivariate quantile and related functions, we provide coherent definitions of these properties. For standardisation of multivariate data to produce equivariance or invariance of procedures, three important types of matrix-valued functional are studied: ‘weak covariance’ (or ‘shape’), ‘transformation–retransformation’ (TR), and ‘strong invariant coordinate system’ (SICS). The clarification of TR affine equivariant versions of the sample spatial quantile function is obtained. It is seen that geometric artefacts of SICS-standardised data are invariant under affine transformation of the original data followed by standardisation using the same SICS functional, subject only to translation and homogeneous scale change. Some applications of SICS standardisation are described.

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  • Robert Serfling, 2010. "Equivariance and invariance properties of multivariate quantile and related functions, and the role of standardisation," Journal of Nonparametric Statistics, Taylor & Francis Journals, vol. 22(7), pages 915-936.
    • Handle: RePEc:taf:gnstxx:v:22:y:2010:i:7:p:915-936
    DOI: 10.1080/10485250903431710
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    Cited by:

    1. M. Shams, 2021. "On weakly equivariant estimators," Statistical Papers, Springer, vol. 62(4), pages 1611-1650, August.
    2. Hamel, Andreas H. & Kostner, Daniel, 2018. "Cone distribution functions and quantiles for multivariate random variables," Journal of Multivariate Analysis, Elsevier, vol. 167(C), pages 97-113.
    3. Serfling, Robert & Wijesuriya, Uditha, 2017. "Depth-based nonparametric description of functional data, with emphasis on use of spatial depth," Computational Statistics & Data Analysis, Elsevier, vol. 105(C), pages 24-45.
    4. Jean-Paul Chavas, 2018. "On multivariate quantile regression analysis," Statistical Methods & Applications, Springer;Società Italiana di Statistica, vol. 27(3), pages 365-384, August.
    5. Marc Hallin & Davy Paindaveine & Miroslav Siman, 2008. "Multivariate quantiles and multiple-output regression quantiles: from L1 optimization to halfspace depth," Working Papers ECARES 2008_042, ULB -- Universite Libre de Bruxelles.
    6. Yi He & John H. J. Einmahl, 2017. "Estimation of extreme depth-based quantile regions," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 79(2), pages 449-461, March.
    7. Ilmonen, Pauliina & Nevalainen, Jaakko & Oja, Hannu, 2010. "Characteristics of multivariate distributions and the invariant coordinate system," Statistics & Probability Letters, Elsevier, vol. 80(23-24), pages 1844-1853, December.
    8. Serfling, Robert & Wang, Yunfei, 2016. "On Liu’s simplicial depth and Randles’ interdirections," Computational Statistics & Data Analysis, Elsevier, vol. 99(C), pages 235-247.
    9. Nordhausen, Klaus & Ruiz-Gazen, Anne, 2021. "On the usage of joint diagonalization in multivariate statistics," TSE Working Papers 21-1268, Toulouse School of Economics (TSE).
    10. Nordhausen, Klaus & Ruiz-Gazen, Anne, 2022. "On the usage of joint diagonalization in multivariate statistics," Journal of Multivariate Analysis, Elsevier, vol. 188(C).
    11. Xin Dang & Hailin Sang & Lauren Weatherall, 2019. "Gini covariance matrix and its affine equivariant version," Statistical Papers, Springer, vol. 60(3), pages 641-666, June.
    12. Ramsay, Kelly & Durocher, Stephane & Leblanc, Alexandre, 2021. "Robustness and asymptotics of the projection median," Journal of Multivariate Analysis, Elsevier, vol. 181(C).
    13. Dai, Wenlin & Genton, Marc G., 2019. "Directional outlyingness for multivariate functional data," Computational Statistics & Data Analysis, Elsevier, vol. 131(C), pages 50-65.
    14. 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.
    15. Klaus Nordhausen & Anne Ruiz-Gazen, 2022. "On the usage of joint diagonalization in multivariate statistics," Post-Print hal-04296111, HAL.
    16. Yves Dominicy & Pauliina Ilmonen & David Veredas, 2017. "Multivariate Hill Estimators," International Statistical Review, International Statistical Institute, vol. 85(1), pages 108-142, April.
    17. Agarwal, Gaurav & Tu, Wei & Sun, Ying & Kong, Linglong, 2022. "Flexible quantile contour estimation for multivariate functional data: Beyond convexity," Computational Statistics & Data Analysis, Elsevier, vol. 168(C).
    18. Davy Paindaveine & Germain Van Bever, 2015. "Discussion of “Multivariate Functional Outlier Detection”, by Mia Hubert, Peter Rousseeuw and Pieter Segaert," Statistical Methods & Applications, Springer;Società Italiana di Statistica, vol. 24(2), pages 223-231, July.
    19. Wang, Shanshan & Serfling, Robert, 2018. "On masking and swamping robustness of leading nonparametric outlier identifiers for multivariate data," Journal of Multivariate Analysis, Elsevier, vol. 166(C), pages 32-49.
    20. Ramsay, Kelly & Durocher, Stéphane & Leblanc, Alexandre, 2019. "Integrated rank-weighted depth," Journal of Multivariate Analysis, Elsevier, vol. 173(C), pages 51-69.
    21. Dominicy, Yves & Heikkilä, Matias & Ilmonen, Pauliina & Veredas, David, 2020. "Flexible multivariate Hill estimators," Journal of Econometrics, Elsevier, vol. 217(2), pages 398-410.

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