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Testing equality between several populations covariance operators

Author

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  • Graciela Boente

    (Universidad de Buenos Aires and IMAS, CONICET)

  • Daniela Rodriguez

    (Universidad de Buenos Aires and IMAS, CONICET)

  • Mariela Sued

    (Universidad de Buenos Aires and CONICET)

Abstract

In many situations, when dealing with several populations, equality of the covariance operators is assumed. An important issue is to study whether this assumption holds before making other inferences. In this paper, we develop a test for comparing covariance operators of several functional data samples. The proposed test is based on the Hilbert–Schmidt norm of the difference between estimated covariance operators. In particular, when dealing with two populations, the test statistic is just the squared norm of the difference between the two covariance operators estimators. The asymptotic behaviour of the test statistic under both the null hypothesis and local alternatives is obtained. The computation of the quantiles of the null asymptotic distribution is not feasible in practice. To overcome this problem, a bootstrap procedure is considered. The performance of the test statistic for small sample sizes is illustrated through a Monte Carlo study and on a real data set.

Suggested Citation

  • Graciela Boente & Daniela Rodriguez & Mariela Sued, 2018. "Testing equality between several populations covariance operators," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 70(4), pages 919-950, August.
    • Handle: RePEc:spr:aistmt:v:70:y:2018:i:4:d:10.1007_s10463-017-0613-1
    DOI: 10.1007/s10463-017-0613-1
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    1. Ferraty, Frederic & Vieu, Philippe & Viguier-Pla, Sylvie, 2007. "Factor-based comparison of groups of curves," Computational Statistics & Data Analysis, Elsevier, vol. 51(10), pages 4903-4910, June.
    2. Davide Pigoli & John A. D. Aston & Ian L. Dryden & Piercesare Secchi, 2014. "Distances and inference for covariance operators," Biometrika, Biometrika Trust, vol. 101(2), pages 409-422.
    3. Stefan Fremdt & Josef G. Steinebach & Lajos Horváth & Piotr Kokoszka, 2013. "Testing the Equality of Covariance Operators in Functional Samples," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 40(1), pages 138-152, March.
    4. Neuhaus, Georg, 1980. "A note on computing the distribution of the norm of Hilbert space valued Gaussian random variables," Journal of Multivariate Analysis, Elsevier, vol. 10(1), pages 19-25, March.
    5. Ferraty, Frederic & Van Keilegom, Ingrid & Vieu, Philippe, 2010. "On the Validity of the Bootstrap in Non-Parametric Functional Regression," LIDAM Reprints ISBA 2010018, Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA).
    6. E. Paparoditis & T. Sapatinas, 2016. "Bootstrap-based testing of equality of mean functions or equality of covariance operators for functional data," Biometrika, Biometrika Trust, vol. 103(3), pages 727-733.
    7. Chang, Chung & Todd Ogden, R., 2009. "Bootstrapping sums of independent but not identically distributed continuous processes with applications to functional data," Journal of Multivariate Analysis, Elsevier, vol. 100(6), pages 1291-1303, July.
    8. Arjun Gupta & Jin Xu, 2006. "On Some Tests of the Covariance Matrix Under General Conditions," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 58(1), pages 101-114, March.
    9. Frédéric Ferraty & Ingrid Van Keilegom & Philippe Vieu, 2010. "On the Validity of the Bootstrap in Non‐Parametric Functional Regression," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 37(2), pages 286-306, June.
    10. Boente, Graciela & Rodriguez, Daniela & Sued, Mariela, 2010. "Inference under functional proportional and common principal component models," Journal of Multivariate Analysis, Elsevier, vol. 101(2), pages 464-475, February.
    11. Dauxois, J. & Pousse, A. & Romain, Y., 1982. "Asymptotic theory for the principal component analysis of a vector random function: Some applications to statistical inference," Journal of Multivariate Analysis, Elsevier, vol. 12(1), pages 136-154, March.
    12. Ferraty, F. & Van Keilegom, I. & Vieu, P., 2012. "Regression when both response and predictor are functions," Journal of Multivariate Analysis, Elsevier, vol. 109(C), pages 10-28.
    13. Ferraty, F. & Van Keilegom, Ingrid & Vieu, P., 2012. "Regression when both response and predictor are functions," LIDAM Reprints ISBA 2012004, Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA).
    14. Boente, Graciela & Fraiman, Ricardo, 2000. "Kernel-based functional principal components," Statistics & Probability Letters, Elsevier, vol. 48(4), pages 335-345, July.
    15. J. Cuesta-Albertos & M. Febrero-Bande, 2010. "A simple multiway ANOVA for functional data," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 19(3), pages 537-557, November.
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    Cited by:

    1. Ahmad, Rauf, 2022. "Tests for proportionality of matrices with large dimension," Journal of Multivariate Analysis, Elsevier, vol. 189(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. Norbert Henze & María Dolores Jiménez‐Gamero, 2021. "A test for Gaussianity in Hilbert spaces via the empirical characteristic functional," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 48(2), pages 406-428, June.
    4. Holger Dette & Kevin Kokot, 2022. "Detecting relevant differences in the covariance operators of functional time series: a sup-norm approach," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 74(2), pages 195-231, April.
    5. Balogoun, Armando Sosthène Kali & Nkiet, Guy Martial & Ogouyandjou, Carlos, 2021. "Asymptotic normality of a generalized maximum mean discrepancy estimator," Statistics & Probability Letters, Elsevier, vol. 169(C).

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