• Bernoulli
  • Volume 19, Number 5B (2013), 2627-2651.

Asymptotics of prediction in functional linear regression with functional outputs

Christophe Crambes and André Mas

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We study prediction in the functional linear model with functional outputs, $Y=SX+\varepsilon$, where the covariates $X$ and $Y$ belong to some functional space and $S$ is a linear operator. We provide the asymptotic mean square prediction error for a random input with exact constants for our estimator which is based on the functional PCA of $X$. As a consequence we derive the optimal choice of the dimension $k_{n}$ of the projection space. The rates we obtain are optimal in minimax sense and generalize those found when the output is real. Our main results hold for class of inputs $X(\cdot )$ that may be either very irregular or very smooth. We also prove a central limit theorem for the predictor. We show that, due to the underlying inverse problem, the bare estimate cannot converge in distribution for the norm of the function space.

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Bernoulli, Volume 19, Number 5B (2013), 2627-2651.

First available in Project Euclid: 3 December 2013

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functional data functional output linear regression model optimality prediction mean square error weak convergence


Crambes, Christophe; Mas, André. Asymptotics of prediction in functional linear regression with functional outputs. Bernoulli 19 (2013), no. 5B, 2627--2651. doi:10.3150/12-BEJ469.

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  • [1] Aguilera, A., Ocaña, F. and Valderrama, M. (2008). Estimation of functional regression models for functional responses by wavelet approximation. In Functional and Operatorial Statistics (S. Dabo-Niang and F. Ferraty, eds.). Contrib. Statist. 15–21. Heidelberg: Springer.
  • [2] Antoch, J., Prchal, L., De Rosa, M. and Sarda, P. (2010). Electricity consumption prediction with functional linear regression using spline estimators. J. Appl. Stat. 37 2027–2041.
  • [3] Araujo, A. and Giné, E. (1980). The Central Limit Theorem for Real and Banach Valued Random Variables. New York: Wiley.
  • [4] Bosq, D. (2000). Linear Processes in Function Spaces: Theory and Applications. Lecture Notes in Statistics 149. New York: Springer.
  • [5] Cardot, H., Crambes, C., Kneip, A. and Sarda, P. (2007). Smoothing splines estimators in functional linear regression with errors-in-variables. Comput. Statist. Data Anal. 51 4832–4848.
  • [6] Cardot, H. and Johannes, J. (2010). Thresholding projection estimators in functional linear models. J. Multivariate Anal. 101 395–408.
  • [7] Cardot, H., Mas, A. and Sarda, P. (2007). CLT in functional linear regression models. Probab. Theory Related Fields 138 325–361.
  • [8] Chiou, J.M., Müller, H.G. and Wang, J.L. (2004). Functional response models. Statist. Sinica 14 675–693.
  • [9] Crambes, C., Kneip, A. and Sarda, P. (2009). Smoothing splines estimators for functional linear regression. Ann. Statist. 37 35–72.
  • [10] Cuevas, A., Febrero, M. and Fraiman, R. (2002). Linear functional regression: The case of fixed design and functional response. Canad. J. Statist. 30 285–300.
  • [11] Dunford, N. and Schwartz, J.T. (1988). Linear Operators, Vols. I & II. New York: Wiley.
  • [12] Engl, H.W., Hanke, M. and Neubauer, A. (1996). Regularization of Inverse Problems. Mathematics and Its Applications 375. Dordrecht: Kluwer Academic.
  • [13] Ferraty, F. and Vieu, P. (2006). Nonparametric Functional Data Analysis: Theory and Practice. Springer Series in Statistics. New York: Springer.
  • [14] Hall, P. and Horowitz, J.L. (2007). Methodology and convergence rates for functional linear regression. Ann. Statist. 35 70–91.
  • [15] Kato, T. (1976). Perturbation Theory for Linear Operators, 2nd ed. Grundlehren der Mathematischen Wissenschaften 132. Berlin: Springer.
  • [16] Lifshits, M.A. (1995). Gaussian Random Functions. Mathematics and Its Applications 322. Dordrecht: Kluwer Academic.
  • [17] Malfait, N. and Ramsay, J.O. (2003). The historical functional linear model. Canad. J. Statist. 31 115–128.
  • [18] Ramsay, J.O. and Dalzell, C.J. (1991). Some tools for functional data analysis. J. R. Stat. Soc. Ser. B Stat. Methodol. 53 539–572. With discussion and a reply by the authors.
  • [19] Ramsay, J.O. and Silverman, B.W. (2005). Functional Data Analysis, 2nd ed. Springer Series in Statistics. New York: Springer.
  • [20] Rudin, W. (1987). Real and Complex Analysis, 3rd ed. New York: McGraw-Hill.
  • [21] Stone, C.J. (1982). Optimal global rates of convergence for nonparametric regression. Ann. Statist. 10 1040–1053.
  • [22] Tikhonov, A.N. and Arsenin, V. Y. (1977). Solutions of Ill-Posed Problems. New York: Wiley.
  • [23] Yao, F., Müller, H.G. and Wang, J.L. (2005). Functional linear regression analysis for longitudinal data. Ann. Statist. 33 2873–2903.