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Dimension reduction in regression

Abstract : In this thesis, we study the problem of dimension reduction through the following regression model Y=g(BX,e), where X is a p dimensional vector, Y belongs to R, the function g is unknown and the noise e is independent of X. We are interested in the estimation of the matrix B, with dimension d times p where d is smaller than p (whose knowledge provides good convergence rates for the estimation of g). This problem is processed according to two different approaches. The first one, called the inverse regression, needs the linearity condition on X. The second one, called semiparametric, do not require such an assumption but only that X has a smooth density. In the context of inverse regression, we focus on two families of methods respectively based on E[X f(Y)] and E[XX^T f(Y)]. For both families, we provide conditions on f that allow an exhaustive estimation of B, and also we compute the better function f by minimizing the asymptotic variance. In the semiparametric context, we give a method for the estimation of the gradient of the regression function. Under some classical semiparametric assumptions, we show the root n consistency of our estimator, the exhaustivity of the estimation and the convergence in the processes space. Within each point, an important question is raised : how to choose the dimension of B ? For this we propose a method that estimates of the rank of a matrix by bootstrap hypothesis testing.
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Submitted on : Tuesday, October 8, 2013 - 4:27:35 PM
Last modification on : Friday, September 23, 2022 - 3:13:45 AM
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  • HAL Id : tel-00871049, version 1


François Portier. Dimension reduction in regression. General Mathematics [math.GM]. Université Rennes 1, 2013. English. ⟨NNT : 2013REN1S039⟩. ⟨tel-00871049⟩



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