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The functional nonparametric model and applications to spectrometric data. (English) Zbl 1037.62032
The goal of this paper is to introduce and study a nonparametric regression model with scalar response when explanatory variables are curves. The problem of regression estimation is reformulated in terms of a general regression model in a semi-normed vector space and kernel smoothing ideas are used. So, the authors construct nonparametric estimates based on convolution kernel ideas and propose a practical automatic procedure that provides an easy computation scheme and deal with the optimality problem. After a simulation study this procedure is applied to real spectrometric data and demonstrates good performance. Finally, asymptotic results with rates of uniform almost sure convergence are presented. Thus, it is shown that the proposed method successfully combines advantages of easy implementation and good mathematical properties.

MSC:
62G08 Nonparametric regression and quantile regression
62G07 Density estimation
Software:
fda (R); KernSmooth; R
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[1] Becker, R., J. Chambers & Wilks, A. (1988), The new S language, a programming environment for data analysis and graphic. Wadsworth and Brooks/Cole. · Zbl 0642.68003
[2] Besse, P.; Cardot, H.; Ferraty, F., Simultaneous non-parametric regressions of unbalanced longitudinal data, Computational Statist and Data Analysis, 24, 225-270 (1997) · Zbl 0900.62199
[3] Besse, P., Cardot, H. & Stephenson, D. (1999), Autoregressive forecasting of some functional climatic variations. Scand. J. of Statist., in print. · Zbl 0962.62089
[4] Boularan, J.; Ferré, L.; Vieu, P., A nonparametric model for unbalanced longitudinal data with application to geophysical data, Computational Statist., 10, 285-298 (1995) · Zbl 0936.62045
[5] Brown, P. J.; Fearn, T.; Vannucci, M., Bayesian Wavelet Regression on Curves With Application to a Spectroscopic Calibration Problem, J. Amer. Statist. Assoc., 96, 398-408 (2001) · Zbl 1022.62027
[6] Cardot, H.; Ferraty, F.; Sarda, P., Functional linear model, Statist. & Proba. Lett., 45, 11-22 (1999) · Zbl 0962.62081
[7] de Boor, C. (1978), A practical Guide to Splines. Springer, New-York. · Zbl 0406.41003
[8] Denham, M. C.; Brown, P. J., Calibration with Many Variables, Appl. Statist., 42, 515-528 (1993) · Zbl 0825.62584
[9] Ferraty, F. & Vieu, P. (2000), Dimension fractale et estimation de la régression dans des espaces vectoriels semi-normés. C. R. Acad. Sci. Paris, 323, 403-406. · Zbl 0942.62045
[10] Ferraty, F. & Vieu, P. (2001), Statistique Fonctionnelle: Modèles de Régression pour Variables Aléatoires Uni, Multi et Infiniment Dimensionnées. Technical Report, Laboratoire de Statistique et Probabilités, Toulouse, 2001-03.
[11] Frank, IE; Friedman, JH, A statistical view of some chemometrics regression tools, Technometrics, 35, 109-148 (1993) · Zbl 0775.62288
[12] Goutis, G.; Fearn, T., Partial Least Squares Regression on Smooth Factors, J. Amer. Statist. Assoc., 91, 627-632 (1996) · Zbl 0869.62051
[13] Härdle, W. (1990), Applied Nonparametric Regression. Cambridge Univ. Press, UK. · Zbl 0714.62030
[14] Hoeffding, W., Probability inequalities for sums of bounded random variables, J. Amer. Statist. Assoc., 58, 15-30 (1963) · Zbl 0127.10602
[15] Kneip, A.; Gasser, T., Statistical tools to analyse data representing a sample of curves, Ann. Statist., 20, 1266-1305 (1992) · Zbl 0785.62042
[16] Leurgans, SE; Moyeed, RA; Silverman, BW, Canonical correlation analysis when the data are curves, J. R. Statist. Soc. B, 55, 725-740 (1993) · Zbl 0803.62049
[17] Mack, YP; Silverman, BW, Weak and strong uniform consistency of kernel regression estimates, Zeit. W.u.v.G., 61, 405-415 (1982) · Zbl 0495.62046
[18] Martens, H. & Naes, T. (1989), Multivariate Calibration, New-York: John Wiley. · Zbl 0732.62109
[19] Nunez-Anton, V.; Rodriguez-Poo, J.; Vieu, P., Longitudinal data with non stationnary errors: a three-stage nonparametric approach, TEST, 8, 201-231 (1999) · Zbl 0945.62042
[20] Ramsay, J.; Dalzell, C., Some tools for functional data analysis, J. R. Statist. Soc. B, 53, 539-572 (1991) · Zbl 0800.62314
[21] Ramsay, J.; Li, X., Curve registration, J. R. Statist. Soc. B, 60, 351-363 (1996) · Zbl 0909.62033
[22] Ramsay, J. & Silverman, B. (1997), Functional Data Analysis. Springer-Verlag. · Zbl 0882.62002
[23] Rice, J.; Silverman, B., Estimating the mean and the covariance structure nonparametrically when the data are curves, J. R. Statist. Soc. B, 53, 233-243 (1991) · Zbl 0800.62214
[24] Schumaker, L. (1981), Spline Functions: Basic Theory. Wiley-Interscience. · Zbl 0449.41004
[25] Stone, C., Optimal global rates of convergence for nonparametric estimators, Ann. Statist., 10, 1040-1053 (1982) · Zbl 0511.62048
[26] Vieu, P., Nonparametric regression: local optimal bandwidth choice, J. R. Statist. Soc. B, 53, 453-474 (1991) · Zbl 0800.62217
[27] Wand, M.P. & Jones, M.C. (1995), Kernel Smoothing, Chapman & Hall, London.
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