Spectral analysis of some random matrices based schemes for stable and robust nonparametric and functional regression estimators

In the first part of this work, we develop and study a random pseudo-inverse based scheme for a stable and robust solution of nonparametric regression problems. For the interval $I=[-1,1],$ we use a random projection over a special orthonormal family of a weighted $L^2(I)-$space, given by the Jacobi normalized polynomials. Then, the pseudo-inverse of this random matrix is used to compute the different expansion coefficients of the nonparametric regression estimator with respect to this orthonormal system. We show that this estimator is stable. Then, we combine the RANdom SAmpling Consensus (RANSAC) iterative algorithm with the previous scheme to get a robust and stable nonparametric regression estimator. This estimator has also the advantage to provide fairly accurate approximations to the true regression functions. In the second part of this work, we extend the random pseudo-inverse scheme technique to build a stable and accurate estimator for solving linear functional regression (LFR) problems. A dyadic decomposition approach is used to construct this last stable estimator for the LFR problem. The performance of the two proposed estimators are illustrated by various numerical simulations.