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Adaptive nonparametric instrumental regression by model selection. arxiv:1003.3128v1, Université Catholique de Louvain
, 2010
"... We consider the problem of estimating the structural function in nonparametric instrumental regression, where in the presence of an instrument W a response Y is modeled in dependence of an endogenous explanatory variable Z. The proposed estimator is based on dimension reduction and additional thres ..."
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We consider the problem of estimating the structural function in nonparametric instrumental regression, where in the presence of an instrument W a response Y is modeled in dependence of an endogenous explanatory variable Z. The proposed estimator is based on dimension reduction and additional thresholding. The minimax optimal rate of convergence of the estimator is derived assuming that the structural function belongs to some ellipsoids which are in a certain sense linked to the conditional expectation operator of Z given W. We illustrate these results by considering classical smoothness assumptions. However, the proposed estimator requires an optimal choice of a dimension parameter depending on certain characteristics of the unknown structural function and the conditional expectation operator of Z given W, which are not known in practice. The main issue addressed in our work is a fully adaptive choice of this dimension parameter using a model selection approach under the restriction that the conditional expectation operator of Z given W is smoothing in a certain sense. In this situation we develop a penalized minimum contrast estimator with randomized penalty and collection of models. We show that this datadriven estimator can attain the lower risk bound up to a constant over a wide range of smoothness classes for the structural function.
Minimax adaptive tests for the Functional Linear model
"... Abstract:. We introduce two novel procedures to test the nullity of the slope function in the functional linear model with real output. The test statistics combine multiple testing ideas and random projections of the input data through functional Principal Component Analysis. Interestingly, the proc ..."
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Abstract:. We introduce two novel procedures to test the nullity of the slope function in the functional linear model with real output. The test statistics combine multiple testing ideas and random projections of the input data through functional Principal Component Analysis. Interestingly, the procedures are completely datadriven and do not require any prior knowledge on the smoothness of the slope nor on the smoothness of the covariate functions. The levels and powers against local alternatives are assessed in a nonasymptotic setting. This allows us to prove that these procedures are minimax adaptive (up to an unavoidable log log n multiplicative term) to the unknown regularity of the slope. As a side result, the minimax separation distances of the slope are derived for a large range of regularity classes. A numerical study illustrates these theoretical results.
Estimation of the sobol indices in a linear functional multidimensional model
 2012. hal00881112, version 1  7 Nov 2013
"... We consider a functional linear model where the explicative variables are stochastic processes taking values in a Hilbert space, the main example is given by Gaussian processes in L2([0, 1]). We propose estimators of the Sobol indices in this functional linear model. Our estimators are based on U−st ..."
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We consider a functional linear model where the explicative variables are stochastic processes taking values in a Hilbert space, the main example is given by Gaussian processes in L2([0, 1]). We propose estimators of the Sobol indices in this functional linear model. Our estimators are based on U−statistics. We prove the asymptotic normality and the efficiency of our estimators and we compare them from a theoretical and practical point of view with classical estimators of Sobol indices. Mathematics Subject Classification:
Nonasymptotic Adaptive Prediction in Functional Linear Models
, 2013
"... 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 contr ..."
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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 nonrandom spaces, we need to adapt it to empirical eigenbasis made of datadependent – 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 crossvalidation.
Adaptive functional linear regression Fabienne Comte∗
"... Université catholique de Louvain 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 [2010] have shown that a thresholded projection estimator can attain up to a constant minimaxr ..."
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Université catholique de Louvain 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 [2010] have shown that a thresholded projection estimator can attain up to a constant minimaxrates of convergence in a general framework which allows 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 datadriven choice of the tuning parameter which combines model selection and Lepski’s method. It is inspired by the recent work of Goldenshluger and Lepski [2011]. 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 datadriven 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.