Non-asymptotic adaptive prediction in functional linear models
| Autor: | Angelina Roche, André Mas, Elodie Brunel |
|---|---|
| Přispěvatelé: | Institut Montpelliérain Alexander Grothendieck (IMAG), Université de Montpellier (UM)-Centre National de la Recherche Scientifique (CNRS) |
| Rok vydání: | 2016 |
| Předmět: |
Statistics and Probability
Mathematical optimization Mathematics - Statistics Theory Statistics Theory (math.ST) 01 natural sciences Projection (linear algebra) Functional linear regression 010104 statistics & probability [MATH.MATH-ST]Mathematics [math]/Statistics [math.ST] Minimax rate FOS: Mathematics Applied mathematics Model selection on random bases 62J05 62G08 0101 mathematics Eigenvalues and eigenvectors Penalized contrast estimator Mathematics Functional principal component analysis Numerical Analysis Model selection 010102 general mathematics Linear model Estimator [STAT.TH]Statistics [stat]/Statistics Theory [stat.TH] Minimax Covariance operator Statistics Probability and Uncertainty Mean squared prediction error |
| Zdroj: | Journal of Multivariate Analysis Journal of Multivariate Analysis, Elsevier, 2016, 143, pp.208-232. ⟨10.1016/j.jmva.2015.09.008⟩ |
| ISSN: | 0047-259X 1095-7243 |
| DOI: | 10.1016/j.jmva.2015.09.008 |
| Popis: | Functional linear regression has recently attracted considerable interest. Many works focus on asymptotic inference. In this paper we consider in a non asymptotic framework a simple estimation procedure based on functional Principal Regression. It revolves in the minimization of a least square contrast coupled with a classical projection on the space spanned by the m first empirical eigenvectors of the covariance operator of the functional sample. The novelty of our approach is to select automatically the crucial dimension m by minimization of a penalized least square contrast. Our method is based on model selection tools. Yet, since this kind of methods consists usually in projecting onto known non-random spaces, we need to adapt it to empirical eigenbasis made of data-dependent - hence random - vectors. The resulting estimator is fully adaptive and is shown to verify an oracle inequality for the risk associated to the prediction error and to attain optimal minimax rates of convergence over a certain class of ellipsoids. Our strategy of model selection is finally compared numerically with cross-validation. |
| Databáze: | OpenAIRE |
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