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Weak convergence in the functional autoregressive model

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  • Mas, André

Abstract

The functional autoregressive model is a Markov model taylored for data of functional nature. It revealed fruitful when attempting to model samples of dependent random curves and has been widely studied along the past few years. This article aims at completing the theoretical study of the model by addressing the issue of weak convergence for estimates from the model. The main difficulties stem from an underlying inverse problem as well as from dependence between the data. Traditional facts about weak convergence in non-parametric models appear: the normalizing sequence is not an , a bias term appears. Several original features of the functional framework are pointed out.

Suggested Citation

  • Mas, André, 2007. "Weak convergence in the functional autoregressive model," Journal of Multivariate Analysis, Elsevier, vol. 98(6), pages 1231-1261, July.
    • Handle: RePEc:eee:jmvana:v:98:y:2007:i:6:p:1231-1261
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    References listed on IDEAS

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    1. André Mas, 1999. "Normalité asymptotique de l’estimateur empirique de l’opérateur d’autocorrélation d’un processus ARH(1)," Working Papers 99-11, Center for Research in Economics and Statistics.
    2. Philippe C. Besse & Herve Cardot & David B. Stephenson, 2000. "Autoregressive Forecasting of Some Functional Climatic Variations," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 27(4), pages 673-687, December.
    3. Mas, André & Menneteau, Ludovic, 2003. "Large and moderate deviations for infinite-dimensional autoregressive processes," Journal of Multivariate Analysis, Elsevier, vol. 87(2), pages 241-260, November.
    4. Antoniadis, Anestis & Sapatinas, Theofanis, 2003. "Wavelet methods for continuous-time prediction using Hilbert-valued autoregressive processes," Journal of Multivariate Analysis, Elsevier, vol. 87(1), pages 133-158, October.
    5. Menneteau, Ludovic, 2005. "Some laws of the iterated logarithm in Hilbertian autoregressive models," Journal of Multivariate Analysis, Elsevier, vol. 92(2), pages 405-425, February.
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    Cited by:

    1. Chi Yao & Wei Yu & Xuejun Wang, 2023. "Strong Consistency for the Conditional Self-weighted M Estimator of GRCA(p) Models," Methodology and Computing in Applied Probability, Springer, vol. 25(1), pages 1-21, March.
    2. Joseph, Esdras & Galeano San Miguel, Pedro & Lillo Rodríguez, Rosa Elvira, 2013. "The Mahalanobis distance for functional data with applications to classification," DES - Working Papers. Statistics and Econometrics. WS ws131312, Universidad Carlos III de Madrid. Departamento de Estadística.
    3. Álvarez-Liébana, J. & Bosq, D. & Ruiz-Medina, M.D., 2017. "Asymptotic properties of a component-wise ARH(1) plug-in predictor," Journal of Multivariate Analysis, Elsevier, vol. 155(C), pages 12-34.
    4. Zhang, Xianyang, 2016. "White noise testing and model diagnostic checking for functional time series," Journal of Econometrics, Elsevier, vol. 194(1), pages 76-95.
    5. Joseph, Esdras & Galeano San Miguel, Pedro & Lillo Rodríguez, Rosa Elvira, 2015. "Two-sample Hotelling's T² statistics based on the functional Mahalanobis semi-distance," DES - Working Papers. Statistics and Econometrics. WS ws1503, Universidad Carlos III de Madrid. Departamento de Estadística.
    6. M. D. Ruiz-Medina & D. Miranda & R. M. Espejo, 2019. "Dynamical multiple regression in function spaces, under kernel regressors, with ARH(1) errors," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 28(3), pages 943-968, September.
    7. Álvarez-Liébana, Javier & Bosq, Denis & Ruiz-Medina, María D., 2016. "Consistency of the plug-in functional predictor of the Ornstein–Uhlenbeck process in Hilbert and Banach spaces," Statistics & Probability Letters, Elsevier, vol. 117(C), pages 12-22.
    8. A. Soltani & M. Hashemi, 2011. "Periodically correlated autoregressive Hilbertian processes," Statistical Inference for Stochastic Processes, Springer, vol. 14(2), pages 177-188, May.
    9. Alexander Gleim & Nazarii Salish, 2022. "Forecasting Environmental Data: An example to ground-level ozone concentration surfaces," Papers 2202.03332, arXiv.org.
    10. Park, Joon Y. & Qian, Junhui, 2012. "Functional regression of continuous state distributions," Journal of Econometrics, Elsevier, vol. 167(2), pages 397-412.
    11. Ruiz-Medina, M.D. & Álvarez-Liébana, J., 2019. "A note on strong-consistency of componentwise ARH(1) predictors," Statistics & Probability Letters, Elsevier, vol. 145(C), pages 224-228.
    12. Caponera, Alessia & Panaretos, Victor M., 2022. "On the rate of convergence for the autocorrelation operator in functional autoregression," Statistics & Probability Letters, Elsevier, vol. 189(C).
    13. Won-Ki Seo & Dakyung Seong, 2025. "Functional Linear Projection and Impulse Response Analysis," Papers 2503.08364, arXiv.org, revised Apr 2025.
    14. Cerovecki, Clément & Hörmann, Siegfried, 2017. "On the CLT for discrete Fourier transforms of functional time series," Journal of Multivariate Analysis, Elsevier, vol. 154(C), pages 282-295.
    15. Xu, Meng & Li, Jialiang & Chen, Ying, 2017. "Varying coefficient functional autoregressive model with application to the U.S. treasuries," Journal of Multivariate Analysis, Elsevier, vol. 159(C), pages 168-183.
    16. A. Berlinet & A. Elamine & A. Mas, 2011. "Local linear regression for functional data," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 63(5), pages 1047-1075, October.

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