Adaptive functional linear regression

By Fabienne Comte, Jan Johannes. 2011

59 citations recorded by the source corpus.

We consider the estimation of the slope function in functional linear regression, where scalar responses are modeled in dependence of random functions. Cardot and Johannes [J. Multivariate Anal. 101 (2010) 395–408] 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 Goldenshluger and Lepski [Ann. Statist. 39 (2011) 1608–1632]. 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 minimaxrates 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.

Subfields: Artificial Intelligence, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/8e79f4f1ffd17c27f58651098060e60c42296116

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Artificial Intelligence

In Statistics and Probability

Citations in the corpus

Truncated linear models for functional data (2014, methodology): “…Yuan and Cai (2010) suggested a method founded on reproducing kernel Hilbert space analysis; Ferraty et al. (2012) discussed presmoothing methods; and Comte and Johannes (2012), Johannes and Schenk (2012) and Cai and Zhou (2013) treated methods for adaptive smoothing in functional linear…”
Adaptive Global Testing for Functional Linear Models (2014, methodology): “A consistent estimator of σ 2 can be obtained by the residual mean square in the linear regression of Y on the first mn estimated principal components, provided that mn grows slowly to infinity as n increases (Cai and Zhou 2008).”
Minimax adaptive tests for the Functional Linear model (2012, background): “Recently, some nonasymptotic results have emerged [11, 12] for estimation procedures that rely on a prescribed basis of functions (e.”