Robust efficient estimation based on model selection for nonparametric regression with semi-Markov noise

In this presentation we consider the nonparametric robust estimation problem for regression models in continuous time with semi-Markov and Lévy noises and we are interested in estimating an unknown function based on observations that can be in continuous or discrete time. This problem of nonparametric estimation in regression models has been considered in many frameworks in the literature. Compared to classical approaches, our objective is to assume that, in addition to the intrinsic noise approximated usually by Brownian motion: (i) the signal is also disturbed by an impulse noise and (ii) we keep the dependence in the noise for a sufficiently large duration.
We construct a series of estimators by projection and thus we approximate the unknown function by a finite Fourier series. We develop an adaptive estimation method based on the model selection procedure proposed by Konev and Pergamenshchikov (2012). First, this procedure gives us a family of estimators; second, we choose the best possible one by minimizing a cost function. Under general moment conditions on the noise distribution, a sharp non-asymptotic oracle inequality for the robust risk is obtained. In particular, we apply the developed model selection methods for the detection problem of the number of signals in multi-path information transmission observed with complex dependent impulse semi-Markov noises.
(Joint work with Slim Beltaief, Serguei Pergamenshchikov; LMRS, Université de Rouen - Normandie)