Nonparametric free deconvolution and circular regression estimation

This thesis contains two independent parts In the first part, we are interested in reconstructing the initial condition of a non-linear partial differential equation (PDE), namely the FokkerPlanck equation, from the observation of a Dyson Brownian motion at a given time $t > 0$. The solution of the Fokker-Planck equation can be written as the free convolution of the initial condition and the semi-circular distribution. We propose a nonparametric estimator for the initial condition obtained by performing the free deconvolution via the subordination functions method. This statistical estimator is original as it involves the resolution of a fixed point equation, and a classical deconvolution by a Cauchy distribution. This is due to the fact that, in free probability, the analogue of the Fourier transform is the R-transform, related to the Cauchy transform. In past literature, there has been a focus on the estimation of the initial conditions of linear PDEs such as the heat equation, but to the best of our knowledge, this is the first time that the problem is tackled for a nonlinear PDE. The convergence of the estimator is proved and the mean integrated squared error is studied, providing rates of convergence similar to the ones known for non-parametric deconvolution methods. Finally, a simulation study illustrates the good performances of our estimator. In the second part of this thesis, the nonparametric estimation for regression with circular responses is studied. We propose a new warped kernel estimator for the regression function at a given point. The bandwidth of kernel is selected by the method of Goldenshluger-Lepski. Under the consideration of pointwise squared risk, we obtain an oracle-type inequality and rates of convergence over Hölder class.