Open Access
November 2013 Asymptotics of prediction in functional linear regression with functional outputs
Christophe Crambes, André Mas
Bernoulli 19(5B): 2627-2651 (November 2013). DOI: 10.3150/12-BEJ469


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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Christophe Crambes. André Mas. "Asymptotics of prediction in functional linear regression with functional outputs." Bernoulli 19 (5B) 2627 - 2651, November 2013.


Published: November 2013
First available in Project Euclid: 3 December 2013

zbMATH: 1280.62084
MathSciNet: MR3160566
Digital Object Identifier: 10.3150/12-BEJ469

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

Rights: Copyright © 2013 Bernoulli Society for Mathematical Statistics and Probability

Vol.19 • No. 5B • November 2013
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