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Title: Estimation in the functional convolution model Authors:  Tito Manrique - UMR MISTEA - INRA Montpellier SUPAGRO (France) [presenting]
Christophe Crambes - University of Montpellier (France)
Nadine Hilgert - UMR MISTEA INRA Montpellier SUPAGRO (France)
Abstract: An estimator is proposed for the unknown function in the Functional Convolution Model, which studies the relationship between a functional covariate $X(t)$ and a functional response $Y(t)$ through the following equation $Y(t)=\int_{0}^t \theta(s) X(t-s) ds + \epsilon(t)$, where $\theta$ is the function to be estimated and $\epsilon$ is an additional functional noise. In this way we can study the influence of the history of $X$ on $Y (t)$. We use the Continuous Fourier Transform to define an estimator of $\theta$. The transformation of the convolution model results in the Functional Concurrent Model associated, in the frequency domain, namely ${\mathcal{Y}}(\xi) = \beta(\xi) {\mathcal{X}}(\xi) + \varepsilon(\xi)$. In order to estimate the unknown function $\beta$, we extended the classical ridge regression method to the functional data framework. We establish consistency properties of the proposed estimators and illustrate our results with some simulations.