Abstract
In this paper, we consider the problem of robust adaptive efficient estimating a periodic signal observed in the transmission channel with the dependent noise defined by non-Gaussian Ornstein-Uhlenbeck processes with unknown correlation properties. Adaptive model selection procedures, based on the shrinkage weighted least squares estimates, are proposed. The comparison between shrinkage and least squares methods is studied and the advantages of the shrinkage methods are analyzed. Estimation properties for proposed statistical algorithms are studied on the basis of the robust mean square accuracy defined as the maximum mean square estimation error over all possible values of unknown noise parameters. Sharp oracle inequalities for the robust risks have been obtained. The robust efficiency of the model selection procedure has been established.
Highlights
We consider the estimation problem for the 1-periodic signal S(t) on the basis of observations0≤t≤n given by the stochastic differential equation: dyt = S(t)dt + dξt, 0 ≤ t ≤ n, (1)
Note that if0≤t≤n is Brownian motion, we obtain the well-known "signal + white noise" model which is very popular in statistical radio-physics
We assume that the useful signal S is distorted by the impulse flow described by the nonGaussian Ornstein-Uhlenbeck processes, which allows studying the signal estimation problems with dependent pulse noises, i.e. we assume that the noise process0≤t≤n obeys the equation: Keywords
Summary
Asymptotic efficiency, model selection, nonparametric regression, Ornstein-Uhlenbeck process, periodic signals, robust quadratic risk, sharp oracle inequality, shrinkage estimation, weighted least squares estimates. The main goal of this paper is to develop a new improved adaptive robust efficient signal estimation methods for the non-Gaussian Ornstein-Uhlenbeck noise (ξt)0≤t≤n based on the general Levy processes with unknown distribution Q. We assume that this distribution belongs to the class Q∗n defined as a family of all these distributions for which the parameters −a∗ ≤ a < 0, n1 ≥ ξ∗ and n21 + n22 ≤ ξ∗, where a∗, ξ∗ and ξ∗ are some fixed positive bounds. This means that the improvement effect in the nonparametric case is more significant than for parametric regression [11]
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