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Comte, Fabienne; Johannes, Jan

Adaptive functional linear regression. (English) Zbl 1373.62350

Ann. Stat. 40, No. 6, 2765-2797 (2012).
Summary: We consider the estimation of the slope function in functional linear regression, where scalar responses are modeled in dependence of random functions. H. Cardot and J. Johannes [J. Multivariate Anal. 101, No. 2, 395–408 (2010; Zbl 1178.62032)] have shown that a thresholded projection estimator can attain up to a constant minimax-rates of convergence in a general framework which allows us to cover the prediction problem with respect to the mean squared prediction error as well as the estimation of the slope function and its derivatives. This estimation procedure, however, requires an optimal choice of a tuning parameter with regard to certain characteristics of the slope function and the covariance operator associated with the functional regressor. As this information is usually inaccessible in practice, we investigate a fully data-driven choice of the tuning parameter which combines model selection and Lepski’s method. It is inspired by the recent work of A. Goldenshluger and O. Lepski [Ann. Stat. 39, No. 3, 1608–1632 (2011; Zbl 1234.62035)]. The tuning parameter is selected as minimizer of a stochastic penalized contrast function imitating Lepski’s method among a random collection of admissible values. This choice of the tuning parameter depends only on the data and we show that within the general framework the resulting data-driven thresholded projection estimator can attain minimax-rates up to a constant over a variety of classes of slope functions and covariance operators. The results are illustrated considering different configurations which cover in particular the prediction problem as well as the estimation of the slope and its derivatives. A simulation study shows the reasonable performance of the fully data-driven estimation procedure.

Cited in 20 Documents

MSC:

62J05 Linear regression; mixed models
62G35 Nonparametric robustness
62G20 Asymptotic properties of nonparametric inference

Keywords:

adaptation; model selestion; Lepski’s method; linear Galerkin approach; prediction; derivative estimation; minimax theory

Citations:

Zbl 1178.62032; Zbl 1234.62035

Software:

fda (R)
Cite Review PDF
Full Text: DOI arXiv Euclid

References:

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