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Thresholding projection estimators in functional linear models
"... We consider the problem of estimating the regression function in functional linear regression models by proposing a new type of projection estimators which combine dimension reduction and thresholding. The introduction of a threshold rule allows to get consistency under broad assumptions as well as ..."
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Cited by 11 (2 self)
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We consider the problem of estimating the regression function in functional linear regression models by proposing a new type of projection estimators which combine dimension reduction and thresholding. The introduction of a threshold rule allows to get consistency under broad assumptions as well as minimax rates of convergence under additional regularity hypotheses. We also consider the particular case of Sobolev spaces generated by the trigonometric basis which permits to get easily estimators of the derivatives of the regression function and prove they are minimax. Rates of convergence are given for some particular cases.
On rate optimal local estimation in functional linear model
, 2009
"... We consider the problem of estimating for a given representer h the value ℓh(β) of a linear functional of the slope parameter β in functional linear regression, where scalar responses Y1,..., Yn are modeled in dependence of random functions X1,..., Xn. The proposed estimators of ℓh(β) are based on d ..."
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Cited by 2 (1 self)
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We consider the problem of estimating for a given representer h the value ℓh(β) of a linear functional of the slope parameter β in functional linear regression, where scalar responses Y1,..., Yn are modeled in dependence of random functions X1,..., Xn. The proposed estimators of ℓh(β) are based on dimension reduction and additional thresholding. The minimax optimal rate of convergence of the estimator is derived assuming that the slope parameter and the representer belong to some ellipsoid which are in a certain sense linked to the covariance operator associated to the regressor. We illustrate these results by considering Sobolev ellipsoids and finitely or infinitely smoothing covariance operator.
hal00873180, version 2 Regularizing Priors for Linear Inverse Problems
"... hal00873180, version 2This paper proposes a new Bayesian approach for estimating, nonparametrically, functional parameters in econometric models that are characterized as the solution of a linear inverse problem. By using a Gaussian process prior distribution we propose the posterior mean as an es ..."
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hal00873180, version 2This paper proposes a new Bayesian approach for estimating, nonparametrically, functional parameters in econometric models that are characterized as the solution of a linear inverse problem. By using a Gaussian process prior distribution we propose the posterior mean as an estimator and provefrequentist consistency ofthe posteriordistribution. The latter providesthe frequentist validation of our Bayesian procedure. We show that the minimax rate of contraction of the posterior distribution can be obtained provided that either the regularity of the prior matches the regularity of the true parameter or the prior is scaled at an appropriate rate. The scaling parameter of the prior distribution plays the role of a regularization parameter. We propose a new datadriven method for optimally selecting in practice this regularization parameter. We also provide sufficient conditions so that the posterior mean, in a conjugateGaussian setting, is equal to a Tikhonovtype estimator in a frequentist setting. Under these conditions our datadriven method is valid for selecting the regularization parameter of the Tikhonov estimator as well. Finally, we apply our general methodology to two leading examples in econometrics: instrumental regression and functional regression estimation.