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A characterization of elliptical distributions and some optimality properties of principal components for functional data

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  • Boente, Graciela
  • Salibián Barrera, Matías
  • Tyler, David E.

Abstract

As in the multivariate setting, the class of elliptical distributions on separable Hilbert spaces serves as an important vehicle and reference point for the development and evaluation of robust methods in functional data analysis. In this paper, we present a simple characterization of elliptical distributions on separable Hilbert spaces, namely we show that the class of elliptical distributions in the infinite-dimensional case is equivalent to the class of scale mixtures of Gaussian distributions on the space. Using this characterization, we establish a stochastic optimality property for the principal component subspaces associated with an elliptically distributed random element, which holds even when second moments do not exist. In addition, when second moments exist, we establish an optimality property regarding unitarily invariant norms of the residuals covariance operator.

Suggested Citation

  • Boente, Graciela & Salibián Barrera, Matías & Tyler, David E., 2014. "A characterization of elliptical distributions and some optimality properties of principal components for functional data," Journal of Multivariate Analysis, Elsevier, vol. 131(C), pages 254-264.
    • Handle: RePEc:eee:jmvana:v:131:y:2014:i:c:p:254-264
    DOI: 10.1016/j.jmva.2014.07.006
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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. N. Locantore & J. Marron & D. Simpson & N. Tripoli & J. Zhang & K. Cohen & Graciela Boente & Ricardo Fraiman & Babette Brumback & Christophe Croux & Jianqing Fan & Alois Kneip & John Marden & Daniel P, 1999. "Robust principal component analysis for functional data," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 8(1), pages 1-73, June.
    3. Daniel Gervini, 2008. "Robust functional estimation using the median and spherical principal components," Biometrika, Biometrika Trust, vol. 95(3), pages 587-600.
    4. Huffer, Fred W. & Park, Cheolyong, 2007. "A test for elliptical symmetry," Journal of Multivariate Analysis, Elsevier, vol. 98(2), pages 256-281, February.
    5. Schott, James R., 2002. "Testing for elliptical symmetry in covariance-matrix-based analyses," Statistics & Probability Letters, Elsevier, vol. 60(4), pages 395-404, December.
    6. Zhu, Li-Xing & Neuhaus, Georg, 2003. "Conditional tests for elliptical symmetry," Journal of Multivariate Analysis, Elsevier, vol. 84(2), pages 284-298, February.
    7. Kuwana, Yoichi & Kariya, Takeaki, 1991. "LBI tests for multivariate normality in exponential power distributions," Journal of Multivariate Analysis, Elsevier, vol. 39(1), pages 117-134, October.
    8. Bali, Juan Lucas & Boente, Graciela, 2009. "Principal points and elliptical distributions from the multivariate setting to the functional case," Statistics & Probability Letters, Elsevier, vol. 79(17), pages 1858-1865, September.
    9. Batsidis, Apostolos & Zografos, Konstantinos, 2013. "A necessary test of fit of specific elliptical distributions based on an estimator of Song’s measure," Journal of Multivariate Analysis, Elsevier, vol. 113(C), pages 91-105.
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    Citations

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    Cited by:

    1. Qiu, Zhiping & Chen, Jianwei & Zhang, Jin-Ting, 2021. "Two-sample tests for multivariate functional data with applications," Computational Statistics & Data Analysis, Elsevier, vol. 157(C).
    2. Boente, Graciela & Rodriguez, Daniela & Sued, Mariela, 2019. "The spatial sign covariance operator: Asymptotic results and applications," Journal of Multivariate Analysis, Elsevier, vol. 170(C), pages 115-128.
    3. Alvarez, Agustín & Boente, Graciela & Kudraszow, Nadia, 2019. "Robust sieve estimators for functional canonical correlation analysis," Journal of Multivariate Analysis, Elsevier, vol. 170(C), pages 46-62.
    4. Italo R. Lima & Guanqun Cao & Nedret Billor, 2019. "Robust simultaneous inference for the mean function of functional data," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 28(3), pages 785-803, September.
    5. Boente, Graciela & Parada, Daniela, 2023. "Robust estimation for functional quadratic regression models," Computational Statistics & Data Analysis, Elsevier, vol. 187(C).
    6. Graciela Boente & Matías Salibian-Barrera, 2015. "S -Estimators for Functional Principal Component Analysis," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 110(511), pages 1100-1111, September.
    7. 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.
    8. Graciela Boente & Matías Salibián-Barrera, 2021. "Robust functional principal components for sparse longitudinal data," METRON, Springer;Sapienza Università di Roma, vol. 79(2), pages 159-188, August.
    9. Dominik Poß & Dominik Liebl & Alois Kneip & Hedwig Eisenbarth & Tor D. Wager & Lisa Feldman Barrett, 2020. "Superconsistent estimation of points of impact in non‐parametric regression with functional predictors," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 82(4), pages 1115-1140, September.
    10. Bali, Juan Lucas & Boente, Graciela, 2015. "Influence function of projection-pursuit principal components for functional data," Journal of Multivariate Analysis, Elsevier, vol. 133(C), pages 173-199.
    11. Fred Espen Benth & Giulia Di Nunno & Dennis Schroers, 2022. "Copula measures and Sklar's theorem in arbitrary dimensions," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 49(3), pages 1144-1183, September.
    12. T. Górecki & Ł. Smaga, 2017. "Multivariate analysis of variance for functional data," Journal of Applied Statistics, Taylor & Francis Journals, vol. 44(12), pages 2172-2189, September.
    13. González–Rodríguez, Gil & Colubi, Ana & González–Manteiga, Wenceslao & Febrero–Bande, Manuel, 2024. "A consistent test of equality of distributions for Hilbert-valued random elements," Journal of Multivariate Analysis, Elsevier, vol. 202(C).
    14. Park, Yeonjoo & Kim, Hyunsung & Lim, Yaeji, 2023. "Functional principal component analysis for partially observed elliptical process," Computational Statistics & Data Analysis, Elsevier, vol. 184(C).

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