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Characteristics of multivariate distributions and the invariant coordinate system

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  • Ilmonen, Pauliina
  • Nevalainen, Jaakko
  • Oja, Hannu

Abstract

We consider a semiparametric multivariate location-scatter model where the standardized random vector of the model is fixed using simultaneously two location vectors and two scatter matrices. The approach using location and scatter functionals based on the first four moments serves as our main example. The four functionals yield in a natural way the corresponding skewness, kurtosis and unmixing matrix functionals. Affine transformation based on the unmixing matrix transforms the variable to an invariant coordinate system. The limiting properties of the skewness, kurtosis, and unmixing matrix estimates are derived under general conditions. We discuss related statistical inference problems, the role of the sample statistics in testing for normality and ellipticity, and connections to invariant coordinate selection and independent component analysis.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:stapro:v:80:y:2010:i:23-24:p:1844-1853
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    References listed on IDEAS

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    1. David E. Tyler & Frank Critchley & Lutz Dümbgen & Hannu Oja, 2009. "Invariant co‐ordinate selection," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 71(3), pages 549-592, June.
    2. Hannu Oja & Davy Paindaveine & Sara Taskinen, 2009. "Parametric and nonparametric test for multivariate independence in IC models," Working Papers ECARES 2009_018, ULB -- Universite Libre de Bruxelles.
    3. 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.
    4. Nordhausen, Klaus & Oja, Hannu & Paindaveine, Davy, 2009. "Signed-rank tests for location in the symmetric independent component model," Journal of Multivariate Analysis, Elsevier, vol. 100(5), pages 821-834, May.
    5. Annaliisa Kankainen & Sara Taskinen & Hannu Oja, 2007. "Tests of multinormality based on location vectors and scatter matrices," Statistical Methods & Applications, Springer;Società Italiana di Statistica, vol. 16(3), pages 357-379, November.
    6. Bera, A. & John, S., 1983. "Tests for multivariate normality with Pearson alternatives," LIDAM Reprints CORE 534, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
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    Cited by:

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    2. Miettinen, Jari & Nordhausen, Klaus & Oja, Hannu & Taskinen, Sara, 2012. "Statistical properties of a blind source separation estimator for stationary time series," Statistics & Probability Letters, Elsevier, vol. 82(11), pages 1865-1873.
    3. François Bachoc & Marc G Genton & Klaus Nordhausen & Anne Ruiz-Gazen & Joni Virta, 2020. "Spatial blind source separation," Biometrika, Biometrika Trust, vol. 107(3), pages 627-646.
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    5. Lee, Seonjoo & Shen, Haipeng & Truong, Young, 2021. "Sampling properties of color Independent Component Analysis," Journal of Multivariate Analysis, Elsevier, vol. 181(C).
    6. Nordhausen, Klaus & Ruiz-Gazen, Anne, 2022. "On the usage of joint diagonalization in multivariate statistics," Journal of Multivariate Analysis, Elsevier, vol. 188(C).
    7. Loperfido, Nicola, 2015. "Vector-valued skewness for model-based clustering," Statistics & Probability Letters, Elsevier, vol. 99(C), pages 230-237.
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    9. Archimbaud, Aurore & Boulfani, Feriel & Gendre, Xavier & Nordhausen, Klaus & Ruiz-Gazen, Anne & Virta, Joni, 2025. "ICS for multivariate functional anomaly detection with applications to predictive maintenance and quality control," Econometrics and Statistics, Elsevier, vol. 33(C), pages 282-303.

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