Bernoulli

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

Asymptotics of prediction in functional linear regression with functional outputs

Christophe Crambes and André Mas

Full-text: Open access

Abstract

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.

Article information

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

Dates
First available in Project Euclid: 3 December 2013

Permanent link to this document
https://projecteuclid.org/euclid.bj/1386078615

Digital Object Identifier
doi:10.3150/12-BEJ469

Mathematical Reviews number (MathSciNet)
MR3160566

Zentralblatt MATH identifier
1280.62084

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

Citation

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. https://projecteuclid.org/euclid.bj/1386078615


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