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General projection-pursuit estimators for the common principal components model: influence functions and Monte Carlo study

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  • Boente, Graciela
  • Pires, Ana M.
  • Rodrigues, Isabel M.

Abstract

The common principal components (CPC) model for several groups of multivariate observations assumes equal principal axes but possibly different variances along these axes among the groups. Under a CPCs model, generalized projection-pursuit estimators are defined by using score functions on the dispersion measure considered. Their partial influence functions are obtained and asymptotic variances are derived from them. When the score function is taken equal to the logarithm, it is shown that, under a proportionality model, the eigenvector estimators are optimal in the sense of minimizing the asymptotic variance of the eigenvectors, for a given scale measure.

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  • Boente, Graciela & Pires, Ana M. & Rodrigues, Isabel M., 2006. "General projection-pursuit estimators for the common principal components model: influence functions and Monte Carlo study," Journal of Multivariate Analysis, Elsevier, vol. 97(1), pages 124-147, January.
    • Handle: RePEc:eee:jmvana:v:97:y:2006:i:1:p:124-147
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    References listed on IDEAS

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    1. Hengjian Cui, 2003. "Asymptotic distributions of principal components based on robust dispersions," Biometrika, Biometrika Trust, vol. 90(4), pages 953-966, December.
    2. Pires, Ana M. & Branco, João A., 2002. "Partial Influence Functions," Journal of Multivariate Analysis, Elsevier, vol. 83(2), pages 451-468, November.
    3. Croux, Christophe, 1994. "Efficient high-breakdown M-estimators of scale," Statistics & Probability Letters, Elsevier, vol. 19(5), pages 371-379, April.
    4. Graciela Boente, 2002. "Influence functions and outlier detection under the common principal components model: A robust approach," Biometrika, Biometrika Trust, vol. 89(4), pages 861-875, December.
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    Cited by:

    1. Boente, Graciela & Pires, Ana M. & Rodrigues, Isabel M., 2010. "Detecting influential observations in principal components and common principal components," Computational Statistics & Data Analysis, Elsevier, vol. 54(12), pages 2967-2975, December.
    2. Trendafilov, Nickolay T., 2010. "Stepwise estimation of common principal components," Computational Statistics & Data Analysis, Elsevier, vol. 54(12), pages 3446-3457, December.
    3. Marc Hallin & Davy Paindaveine & Thomas Verdebout, 2014. "Efficient R-Estimation of Principal and Common Principal Components," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 109(507), pages 1071-1083, September.
    4. Graciela Boente & Frank Critchley & Liliana Orellana, 2007. "Influence functions of two families of robust estimators under proportional scatter matrices," Statistical Methods & Applications, Springer;Società Italiana di Statistica, vol. 15(3), pages 295-327, February.
    5. Boente, Graciela & Molina, Julieta & Sued, Mariela, 2010. "On the asymptotic behavior of general projection-pursuit estimators under the common principal components model," Statistics & Probability Letters, Elsevier, vol. 80(3-4), pages 228-235, February.
    6. Graciela Boente & Frank Critchley & Liliana Orellana, 2007. "Influence functions of two families of robust estimators under proportional scatter matrices," Statistical Methods & Applications, Springer;Società Italiana di Statistica, vol. 15(3), pages 295-327, February.
    7. Bianco, Ana & Boente, Graciela & Pires, Ana M. & Rodrigues, Isabel M., 2008. "Robust discrimination under a hierarchy on the scatter matrices," Journal of Multivariate Analysis, Elsevier, vol. 99(6), pages 1332-1357, July.
    8. Luca Bagnato & Antonio Punzo, 2021. "Unconstrained representation of orthogonal matrices with application to common principal components," Computational Statistics, Springer, vol. 36(2), pages 1177-1195, June.
    9. Paindaveine, Davy & Rasoafaraniaina, Rondrotiana Joséa & Verdebout, Thomas, 2017. "Preliminary test estimation for multi-sample principal components," Econometrics and Statistics, Elsevier, vol. 2(C), pages 106-116.
    10. Bali, Juan Lucas & Boente, Graciela, 2017. "Robust estimators under a functional common principal components model," Computational Statistics & Data Analysis, Elsevier, vol. 113(C), pages 424-440.

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