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Weak convergence in the functional autoregressive model

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  • Mas, André
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    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.

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    File URL: http://www.sciencedirect.com/science/article/pii/S0047-259X(06)00090-X
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    Article provided by Elsevier in its journal Journal of Multivariate Analysis.

    Volume (Year): 98 (2007)
    Issue (Month): 6 (July)
    Pages: 1231-1261

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    Handle: RePEc:eee:jmvana:v:98:y:2007:i:6:p:1231-1261
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    Keywords: Functional data Autoregressive model Hilbert space Weak convergence Random operator Perturbation theory Linear inverse problem Martingale difference arrays;

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    1. Philippe C. Besse, 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.
    2. 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.
    3. André Mas, 1999. "Normalité asymptotique de l’estimateur empirique de l’opérateur d’autocorrélation d’un processus ARH(1)," Working Papers 99-11, Centre de Recherche en Economie et Statistique.
    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. Lillo, Rosa E. & Galeano, Pedro & Joseph, Esdras, 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.
    2. 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.
    3. Zhang, Xianyang, 2016. "White noise testing and model diagnostic checking for functional time series," Journal of Econometrics, Elsevier, vol. 194(1), pages 76-95.
    4. Lillo, Rosa E. & Galeano, Pedro & Joseph, Esdras, 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.
    5. 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.
    6. A. Soltani & M. Hashemi, 2011. "Periodically correlated autoregressive Hilbertian processes," Statistical Inference for Stochastic Processes, Springer, vol. 14(2), pages 177-188, May.
    7. Park, Joon Y. & Qian, Junhui, 2012. "Functional regression of continuous state distributions," Journal of Econometrics, Elsevier, vol. 167(2), pages 397-412.
    8. 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.
    9. 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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