Projection estimation for inverses problems on Laguerre and Hermite spaces

Overview & Synopsis

In this thesis, we develop non parametric estimation procedures for various inverse problems on Laguerre and Hermite classes. The first part is dedicated to the estimation of the derivatives of a density function in the Laguerre and Hermite basis. The second part is devoted to the estimation of a density function and a regression function in the Hermite basis in a convolution model. Different estimation methods are presented : projection method based on a development of the function of interest (density, derivative of a density, regression function) in the Laguerre or Hermite basis ; mixed deconvolution-projection based on a development in the Hermite basis and an inverse Fourier transformation. For each of these methods, we establish bounds for quadratic risk. For adequate choices of parameters (dimension of projection space or cut-off), we obtain rates of convergence of our estimators. However, theses parameters depend on unknown quantities. Thus, we propose adaptive procedures to select them in a relevant way inspired by model selection by penalization of Birgé and Massart (1997)'s type or Goldenshluger and Lepski (2011)'s methods criterions and we prove non-asymptotic oracle inequalities using concentration inequalities. Numerical studies and comparisons with other strategies are presented to illustrate the good performance of the proposed methods.