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Nonparametric models for functional data, with application in regression, time-series prediction and curve discrimination. (English) Zbl 1049.62039
Summary: The aim of this article is to investigate a new approach for estimating a regression model with scalar response and in which the explanatory variable is valued in some abstract semi-metric functional space. Nonparametric estimates are introduced, and their behaviors are investigated in the situation of dependent data. Our study contains asymptotic results with rates. The curse of dimensionality, which is of great importance in this infinite-dimensional setting, is highlighted by our asymptotic results. Some ideas, based on fractal dimension modelizations, are given to reduce dimensionality of the problem.
Generalization of the model leads to possible applications in several fields of applied statistics, and we present three applications among these, namely: regression estimation, time-series prediction, and curve discrimination. As a by-product of our approach in the finite-dimensional context, we give a new proof for the rates of convergence of some Nadaraya-Watson kernel-type smoothers without needing any smoothness assumption on the density function of the explanatory variables.

MSC:
62G08 Nonparametric regression and quantile regression
62G20 Asymptotic properties of nonparametric inference
62G07 Density estimation
Software:
fda (R)
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References:
[1] Aneiros-Perez G., Proceedings of TIES 2002 Conference (2002)
[2] Berlinet A., Asymptotics in Statistics and Probability (2001)
[3] Bosq D., Nonparametric Statistics for Stochastic Process. Estimation and Prediction 110 (1998) · Zbl 0902.62099
[4] Bosq D., Linear Process in Function Spaces 149 (2000) · Zbl 0962.60004
[5] DOI: 10.1111/1467-9469.00329 · Zbl 1034.62037
[6] DOI: 10.2307/3315952 · Zbl 1012.62039
[7] Ferraty F., Proceedings of SFC 2001 Conference (2001)
[8] DOI: 10.1007/BF02595710 · Zbl 1020.62089
[9] Ferraty F., Com. Rend. Acad. Sci. Paris 330 pp 403– (2000)
[10] DOI: 10.1007/s001800200126 · Zbl 1037.62032
[11] DOI: 10.1016/S0167-9473(03)00032-X · Zbl 1429.62241
[12] DOI: 10.1111/1467-9868.00148 · Zbl 0909.62030
[13] DOI: 10.2307/1269658
[14] Niang S., Comp. Rend. Acad. Sci. Paris Ser. 1 334 pp 213– (2002)
[15] Ramsay J., Functional Data Analysis (1997) · Zbl 0882.62002
[16] Rio E., SMAI, Mathématiques Applications 31 (1999)
[17] Sarda P., Smoothing and Regression: Approaches, Computation and Application (2000)
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.