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Test of independence for functional data

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  • Horváth, Lajos
  • Hušková, Marie
  • Rice, Gregory

Abstract

We wish to test the null hypothesis that a collection of functional observations are independent and identically distributed. Our procedure is based on the sum of the L2 norms of the empirical correlation functions. The limit distribution of the proposed test statistic is established under the null hypothesis. Under the alternative the sample exhibits serial correlation, and consistency is shown when the sample size as well as the number of lags used in the test statistic tend to ∞. A Monte Carlo study illustrates the small sample behavior of the test and the procedure is applied to data sets, Eurodollar futures and magnetogram records.

Suggested Citation

  • Horváth, Lajos & Hušková, Marie & Rice, Gregory, 2013. "Test of independence for functional data," Journal of Multivariate Analysis, Elsevier, vol. 117(C), pages 100-119.
    • Handle: RePEc:eee:jmvana:v:117:y:2013:i:c:p:100-119
    DOI: 10.1016/j.jmva.2013.02.005
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    References listed on IDEAS

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    6. Esam Mahdi & A. Ian McLeod, 2012. "Improved multivariate portmanteau test," Journal of Time Series Analysis, Wiley Blackwell, vol. 33(2), pages 211-222, March.
    7. Gabrys, Robertas & Horváth, Lajos & Kokoszka, Piotr, 2010. "Tests for Error Correlation in the Functional Linear Model," Journal of the American Statistical Association, American Statistical Association, vol. 105(491), pages 1113-1125.
    8. Mas, André, 2002. "Weak convergence for the covariance operators of a Hilbertian linear process," Stochastic Processes and their Applications, Elsevier, vol. 99(1), pages 117-135, May.
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    11. Kargin, V. & Onatski, A., 2008. "Curve forecasting by functional autoregression," Journal of Multivariate Analysis, Elsevier, vol. 99(10), pages 2508-2526, November.
    12. Gabrys, Robertas & Kokoszka, Piotr, 2007. "Portmanteau Test of Independence for Functional Observations," Journal of the American Statistical Association, American Statistical Association, vol. 102, pages 1338-1348, December.
    13. Berkes, István & Horváth, Lajos & Rice, Gregory, 2013. "Weak invariance principles for sums of dependent random functions," Stochastic Processes and their Applications, Elsevier, vol. 123(2), pages 385-403.
    14. Canal, Luisa, 2005. "A normal approximation for the chi-square distribution," Computational Statistics & Data Analysis, Elsevier, vol. 48(4), pages 803-808, April.
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    Citations

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    Cited by:

    1. Zhang, Xianyang, 2016. "White noise testing and model diagnostic checking for functional time series," Journal of Econometrics, Elsevier, vol. 194(1), pages 76-95.
    2. Valentina Masarotto & Victor M. Panaretos & Yoav Zemel, 2019. "Procrustes Metrics on Covariance Operators and Optimal Transportation of Gaussian Processes," Sankhya A: The Indian Journal of Statistics, Springer;Indian Statistical Institute, vol. 81(1), pages 172-213, February.
    3. Zdeněk Hlávka & Marie Hušková & Simos G. Meintanis, 2021. "Testing serial independence with functional data," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 30(3), pages 603-629, September.
    4. Mestre, Guillermo & Portela, José & Rice, Gregory & Muñoz San Roque, Antonio & Alonso, Estrella, 2021. "Functional time series model identification and diagnosis by means of auto- and partial autocorrelation analysis," Computational Statistics & Data Analysis, Elsevier, vol. 155(C).
    5. Atefeh Zamani & Hossein Haghbin & Maryam Hashemi & Rob J. Hyndman, 2022. "Seasonal functional autoregressive models," Journal of Time Series Analysis, Wiley Blackwell, vol. 43(2), pages 197-218, March.
    6. Rice, Gregory & Wirjanto, Tony & Zhao, Yuqian, 2019. "Tests for conditional heteroscedasticity with functional data and goodness-of-fit tests for FGARCH models," MPRA Paper 93048, University Library of Munich, Germany.
    7. Petrovich, Justin & Reimherr, Matthew, 2017. "Asymptotic properties of principal component projections with repeated eigenvalues," Statistics & Probability Letters, Elsevier, vol. 130(C), pages 42-48.
    8. Axel Bücher & Holger Dette & Florian Heinrichs, 2023. "A portmanteau-type test for detecting serial correlation in locally stationary functional time series," Statistical Inference for Stochastic Processes, Springer, vol. 26(2), pages 255-278, July.
    9. Gregory Rice & Tony Wirjanto & Yuqian Zhao, 2020. "Tests for conditional heteroscedasticity of functional data," Journal of Time Series Analysis, Wiley Blackwell, vol. 41(6), pages 733-758, November.
    10. Horváth, Lajos & Liu, Zhenya & Rice, Gregory & Wang, Shixuan, 2020. "A functional time series analysis of forward curves derived from commodity futures," International Journal of Forecasting, Elsevier, vol. 36(2), pages 646-665.
    11. Horváth, Lajos & Rice, Gregory, 2015. "Testing for independence between functional time series," Journal of Econometrics, Elsevier, vol. 189(2), pages 371-382.
    12. Hui Ding & Mei Yao & Riquan Zhang, 2023. "A new estimation in functional linear concurrent model with covariate dependent and noise contamination," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 86(8), pages 965-989, November.
    13. Meintanis, Simos G. & Hušková, Marie & Hlávka, Zdeněk, 2022. "Fourier-type tests of mutual independence between functional time series," Journal of Multivariate Analysis, Elsevier, vol. 189(C).
    14. Buddhananda Banerjee & Satyaki Mazumder, 2018. "A more powerful test identifying the change in mean of functional data," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 70(3), pages 691-715, June.
    15. Kokoszka, Piotr & Reimherr, Matthew & Wölfing, Nikolas, 2016. "A randomness test for functional panels," Journal of Multivariate Analysis, Elsevier, vol. 151(C), pages 37-53.

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