Nonparametric kernel regression estimation for functional stationary ergodic data: Asymptotic properties

Journal of Multivariate Analysis

Volumn 101, Issue 10, 2010, Pages 2266-2281

  Laib, Naâmane ^a   Louani, Djamal ^a  

^a University of Paris VII LPSM partially supported by ERC Starting Grant 680275 MALIG   (France)

Author keywords

Asymptotic normality; Consistency; Ergodic processes; Functional dependent data; Martingale difference; Regression estimation

Indexed keywords

EID: 77956724405     PISSN: 0047259X     EISSN: 10957243     Source Type: Journal    
DOI: 10.1016/j.jmva.2010.05.010     Document Type: Article

References (24)

1. Andrews D.W.K. Non-strong mixing autoregressive processes. J. Appl. Probab. 1984, 21:930-934.
2. L. Aspirot, K. Bertin, G. Perera, Asymptotic normality of the Nadaraya-Watson estimator for non-stationary functional data and applications to telecommunications, Preprint, 2007.
3. Bosq D. Nonparametric Statistics for Stochastic Processes. Lecture Note in Statistics 1998, Springer, New York.
4. Carbon M., Tran L.T. On histograms of linear processes. J. Statist. Plann. Inference 1996, 53:403-419.
5. Chernick M.R. A limit theorem for the maximum of autoregressive processes with uniform marginal distributions. Ann. Probab. 1981, 9:145-149.
6. Chow Y.S., Teicher H. Probabilty Theory 1998, Springer, New York. 2nd ed.
7. Delsol L. Advances on asymptotic normality in non-parametric functional time series analysis. Statistics 2009, 43(1):13-33.
8. Ezzahrioui M., Ould-Saïd E. Asymptotic normality of a nonparametric estimator of the conditional mode function for functional data. J. Nonparametr. Stat. 2008, 20:3-18.
9. Ferraty F., Mas A., Vieu P. Nonparametric regression of functional data: inference and practical aspects. Aust. N. Z. J. Stat. 2007, 49:267-286.
10. Ferraty F., Vieu P. Dimension fractale et estimation de la régression dans des espaces vectoriels semi-normés. C. R. Acad. Sci. Paris Sér. I Math. 2000, 330:139-142.
11. Ferraty F., Vieu P. Nonparametric models for functional data, with applications in regression, time series prediction and curve discrimination. J. Nonparametr. Stat. 2004, 16:111-125.
12. Ferraty F., Vieu P. Nonparametric Functional Data Analysis. Theory and Practice. Springer Series in Statistics 2006, Springer, New York.
13. Gasser T., Hall P., Presnell B. Nonparametric estimation of the mode of a distribution of random curves. J. R. Stat. Soc. Ser. B 1998, 60:681-691.
14. Györfi L., Morvai G., Yakowitz S. Limits to consistent on line forecasting for ergodic time series. IEEE Trans. Inform. Theory 1998, 44:868-892.
15. Hall P., Heyde C. Martingale Limit Theory and its Application 1980, Academic Press, New York.
16. Krengel U. Ergodic Theorems 1985, Walter de Gruyter & Co., Berlin.
17. Laïb N. Kernel estimates of the mean and the volatility functions in a nonlinear autoregressive model with ARCH errors. J. Statist. Plann. Inference 2005, 134:116-139.
18. N. Laïb, D. Louani, Strong consistency of the regression function estimator for functional stationary ergodic data, J. Statist. Plann. Inference (2010), in press.
19. Masry E. Nonparametric regression estimation for dependent functional data: asymptotic normality. Stochastic Process. Appl. 2005, 115:155-177.
20. Politis D.N., Romano J.P. Limit theorems for weakly dependent hilbert space valued random variables with application to the stationary bootstrap. Statist. Sinica 1994, 4:461-476.
21. D.N. Politis, H. White, Automatic block-length selection for the dependent bootstrap, Econometric Reviews, UCSD Manuscript, 2004.
22. Ramsay J., Silverman B.W. Applied Functional Data Analysis: Methods and Case Studies 2002, Springer, New York.
23. Ramsay J., Silverman B.W. Functional Data Analysis 1997, Springer, New York.
24. Tran L.T. Density estimation for time series by histograms. J. Statist. Plann. Inference 1994, 40:61-79.