×

Functional linear model. (English) Zbl 0962.62081

Summary: We study a regression model in which explanatory variables are sampling points of a continuous-time process. We propose an estimator of regression by means of a functional principal components analysis analogous to the one introduced by D. Bosq [Nonparametric Functional Estimation and Related Topics, NATO ASI Ser. Ser. C 335, 509-529 (1991; Zbl 0737.62032)] in the case of Hilbertian AR processes. Both convergence in probability and almost sure convergence of this estimator are stated.

MSC:

62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH)
62J99 Linear inference, regression
62H25 Factor analysis and principal components; correspondence analysis

Citations:

Zbl 0737.62032

Software:

fda (R)
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Ash, R.B., Gardner, M.F., 1975. Topics in Stochastic Processes. Academic Press, New York.; Ash, R.B., Gardner, M.F., 1975. Topics in Stochastic Processes. Academic Press, New York. · Zbl 0317.60014
[2] Besse, P.; Cardot, H., Approximation spline de la prévision d’un processus fonctionnel autorégressif d’ordre 1, Revue Canadienne de Statistique/Canad. J. Statist., 24, 467-487 (1996) · Zbl 0879.62092
[3] Bosq, D., 1991. Modelization, non-parametric estimation and prediction for continuous time processes. In: Roussas, G. (Ed.), Nonparametric Functional Estimation and Related Topics, NATO, ASI Series, pp. 509-529.; Bosq, D., 1991. Modelization, non-parametric estimation and prediction for continuous time processes. In: Roussas, G. (Ed.), Nonparametric Functional Estimation and Related Topics, NATO, ASI Series, pp. 509-529. · Zbl 0737.62032
[4] Cardot, H., 1998. Convergence du lissage spline de la prévision des processus autorégressifs fonctionnels. C.R. Acad. Sci. Paris, Sér. I, t. 326, 755-758.; Cardot, H., 1998. Convergence du lissage spline de la prévision des processus autorégressifs fonctionnels. C.R. Acad. Sci. Paris, Sér. I, t. 326, 755-758. · Zbl 0942.62112
[5] Cardot, H., Ferraty, F., Sarda, P., 1998. Modèle linéaire fonctionnel. Publ. Lab. Statist. Probab. 04:98, Toulouse, France.; Cardot, H., Ferraty, F., Sarda, P., 1998. Modèle linéaire fonctionnel. Publ. Lab. Statist. Probab. 04:98, Toulouse, France.
[6] Dauxois, J.; Pousse, A.; Romain, Y., Asymptotic theory for the principal component analysis of a random vector function: some applications to statistical inference, J. Multivariate Anal., 12, 136-154 (1982) · Zbl 0539.62064
[7] Frank, I. E.; Friedman, J. H., A statistical view of some chemometrics regression tools, Technometrics, 35, 109-148 (1993) · Zbl 0775.62288
[8] Grenander, U., 1963. Probabilities on Algebraic Structures. Almqvist & Wiksell, Stockholm.; Grenander, U., 1963. Probabilities on Algebraic Structures. Almqvist & Wiksell, Stockholm.
[9] Hastie, T.; Buja, A.; Tibshirani, R., Penalized discriminant analysis, Ann. Statist., 23, 73-102 (1995) · Zbl 0821.62031
[10] Hastie, T.; Mallows, C., A discussion of “A Statistical View of Some Chemometrics Regression Tools” by I.E. Frank and J.H. Friedman, Technometrics, 35, 140-143 (1993)
[11] Marx, B.D., Eilers, P.H., 1996. Generalized linear regression on sampled signals with penalized likelihood. In: Forcina, A., Marchetti, G.M., Hatzinger, R., Galmacci, G. (Eds.), Statistical Modelling, Proceedings of the Eleventh International Workshop on Statistical Modelling, Orvietto.; Marx, B.D., Eilers, P.H., 1996. Generalized linear regression on sampled signals with penalized likelihood. In: Forcina, A., Marchetti, G.M., Hatzinger, R., Galmacci, G. (Eds.), Statistical Modelling, Proceedings of the Eleventh International Workshop on Statistical Modelling, Orvietto.
[12] Ramsay, J.O., Silverman, B.W., 1997. Functional Data Analysis. Springer, Berlin.; Ramsay, J.O., Silverman, B.W., 1997. Functional Data Analysis. Springer, Berlin. · Zbl 0882.62002
[13] Vieu, P., Order choice in nonlinear autoregressive models, Statistics, 26, 307-328 (1995) · Zbl 0836.62067
[14] Yurinskiı̆, V. V., Exponential inequalities for sums of random vectors, J. Multivariate Anal., 6, 473-499 (1976) · Zbl 0346.60001
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.