×

Nonparametric time series prediction: A semi-functional partial linear modeling. (English) Zbl 1133.62075

Summary: There is a recent interest in developing new statistical methods to predict time series by taking into account a continuous set of past values as predictors. In this functional time series prediction approach, we propose a functional version of the partial linear model that allows both to consider additional covariates and to use a continuous path in the past to predict future values of the process. The aim of this paper is to present this model, to construct some estimates and to look at their properties both from a theoretical point of view by means of asymptotic results and from a practical perspective by treating some real data sets. Although the literature on the use of parametric or nonparametric functional modeling is growing, as far as we know, this is the first paper on semiparametric functional modeling for the prediction of time series.

MSC:

62M20 Inference from stochastic processes and prediction
62G20 Asymptotic properties of nonparametric inference
62G08 Nonparametric regression and quantile regression
62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH)
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Aneiros-Pérez, G.; González-Manteiga, W.; Vieu, P., Estimation and testing in a partial regression model under long-memory dependence, Bernoulli, 10, 49-78 (2004) · Zbl 1040.62028
[2] Aneiros-Pérez, G.; Vieu, P., Semi-functional partial linear regression, Statist. Probab. Lett., 76, 1102-1110 (2006) · Zbl 1090.62036
[3] Avramidis, P., Two-step cross-validation selection method for partially linear models, Statist. Sinica, 15, 1033-1048 (2005) · Zbl 1086.62050
[4] D. Bosq, Nonparametric Statistics for Stochastic Processes: Estimation and Prediction, second ed., Lecture Notes in Statistics, vol. 110, Springer, Berlin, 1998.; D. Bosq, Nonparametric Statistics for Stochastic Processes: Estimation and Prediction, second ed., Lecture Notes in Statistics, vol. 110, Springer, Berlin, 1998. · Zbl 0902.62099
[5] D. Bosq, Linear Processes in Function Spaces. Estimation and Prediction, Lecture Notes in Statistics, vol. 149, Springer, Berlin, 2000.; D. Bosq, Linear Processes in Function Spaces. Estimation and Prediction, Lecture Notes in Statistics, vol. 149, Springer, Berlin, 2000. · Zbl 0962.60004
[6] Engle, R.; Granger, C.; Rice, J.; Weiss, A., Nonparametric estimates of the relation between weather and electricity sales, J. Amer. Statist. Assoc., 81, 310-320 (1986)
[7] Fan, J.; Yao, Q., Nonlinear Time Series. Nonparametric and Parametric Methods, Springer Series in Statistics (2003), Springer: Springer New York · Zbl 1014.62103
[8] Ferraty, F.; Goia, A.; Vieu, P., Functional nonparametric model for time series: a fractal approach to dimension reduction, Test, 11, 317-344 (2002) · Zbl 1020.62089
[9] F. Ferraty, A. Mas, P. Vieu, Nonparametric regression on functional data: inference and practical aspects, Aust. N.Z. J. Statist. (2007), 10.1111/j.1467-842X.2006.00467.x; F. Ferraty, A. Mas, P. Vieu, Nonparametric regression on functional data: inference and practical aspects, Aust. N.Z. J. Statist. (2007), 10.1111/j.1467-842X.2006.00467.x · Zbl 1136.62031
[10] Ferraty, F.; Vieu, P., Nonparametric models for functional data, with applications in regression, curves discrimination and time series prediction, J. Nonparametric Statist., 16, 111-127 (2004) · Zbl 1049.62039
[11] Ferraty, F.; Vieu, P., Nonparametric Functional Data Analysis, Springer Series in Statistics (2006), Springer: Springer New York · Zbl 1119.62046
[12] Ibragimov, I. A., Some limit theorems for stationary processes, Theoret. Probab. Appl., 7, 282-349 (1962) · Zbl 0119.14204
[13] Masry, E., Nonparametric regression estimation for dependent functional data: asymptotic normality, Stochastic Process. Appl., 115, 155-177 (2005) · Zbl 1101.62031
[14] Oodaira, H.; Yoshihara, K. I., The law of the iterated logarithm for stationary processes satisfying mixing conditions, Kodai Math. Sem. Rep., 23, 311-334 (1971) · Zbl 0234.60034
[15] Speckman, P., Kernel smoothing in partial linear models, J. Roy. Statist. Soc. Ser. B, 50, 413-436 (1988) · Zbl 0671.62045
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.