# zbMATH — the first resource for mathematics

Advances on asymptotic normality in non-parametric functional time series analysis. (English) Zbl 1278.62052
Summary: We consider a stationary process and wish to predict future values from previous ones. Instead of considering the process in its discretized form, we choose to see it as a sample of dependent curves. Then, we cut the process into $$N$$ successive curves. Obviously, the $$N$$ curves are not independent. The prediction issue can be translated into a non-parametric functional regression problem from dependent functional variables. This paper aims to revisit and complete two recent works on this topic. This article extends recent literature and provides asymptotic law with explicit constants under $$\alpha$$-mixing assumptions. Then we establish pointwise confidence bands for the regression function. To conclude, we present how our results behave on a simulation and on a real time series.

##### MSC:
 62G08 Nonparametric regression and quantile regression 60G10 Stationary stochastic processes 60G25 Prediction theory (aspects of stochastic processes) 62G20 Asymptotic properties of nonparametric inference 62M20 Inference from stochastic processes and prediction
fda (R)
Full Text:
##### References:
 [1] Ramsay, J. and Silverman, B. W. 1997. ”Functional Data Analysis”. New York: Springer-Verlag. · Zbl 0882.62002 [2] Ramsay, J. and Silverman, B. W. 2002. ”Applied functional data analysis: Methods and case studies”. New York: Springer-Verlag. · Zbl 1011.62002 [3] Davidian M., Statist. Sinica 14 pp 613– (2004) [4] DOI: 10.1016/j.csda.2006.10.017 · Zbl 1162.62338 [5] Ferraty, F. and Vieu, P. 2006. ”Nonparametric Modelling for Functional Data”. New York: Springer-Verlag. · Zbl 1119.62046 [6] DOI: 10.1214/aos/1176345969 · Zbl 0511.62048 [7] Bosq, D. ”Modelization, Nonparametric Estimation and Prediction for Continuous Time Proccesses”. Edited by: Roussas, G. Vol. 335, 509–530. Nonparametric functional estimation and related topics, NATO ASI Series, Kluwer · Zbl 0737.62032 [8] Bosq D., Lecture Notes in Statistics 149 (2000) [9] DOI: 10.1016/j.spa.2004.07.006 · Zbl 1101.62031 [10] DOI: 10.1111/j.1467-842X.2007.00480.x · Zbl 1136.62031 [11] DOI: 10.1051/ps:1997102 · Zbl 0869.60021 [12] Rio, E. 2000. ”Théorie Asymptotique Des Processus Aléatoires Faiblement Dépendants, Mathématiques et applications”. Vol. 31, Berlin: Springer-Verlag. [13] Bosq, D. 1996. ”Nonparametric Statistics for Stochastic Processes, Estimation and Prediction”. Vol. 110, New York: Springer-Verlag. Lecture Notes in Statistics · Zbl 0857.62081 [14] Liebscher E., Math. Methods Stat 10 pp 194– (2001) [15] DOI: 10.1073/pnas.42.1.43 · Zbl 0070.13804 [16] Ramsay J. O., J. Roy. Statist. Soc. (Ser. B) 53 pp 539– (1991) [17] DOI: 10.1214/ss/1177013604 [18] Cardot, H., Crambes, C. and Sarda, P. 2007. ”Ozone pollution forecasting using conditional mean and conditional quantiles with functional covariates, Statistical Methods for Biostatistics and related fields”. 221–243. Berlin, Heidelberg: Springer. [19] DOI: 10.1016/j.jspi.2006.10.001 · Zbl 1331.62240
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.