Nonparametric Estimation of Transfer Functions: Rates of Convergence and Adaptation

The paper deals with estimating the transfer functions of stable linear time-invariant systems under stochastic assumptions. We adopt a nonparametric minimax approach for measuring the estimation accuracy. The quality of an estimator is measured by its worst case error over a family of transfer functions. The families with polynomially and exponentially decaying impulse response sequences are considered. We establish nonasymptotic upper bounds on the accuracy of the least squares estimator for finite impulse response approximation. It is shown that the attainable estimation accuracy is determined essentially by the rate at which the "true" impulse response tends to zero. Lower bounds on the estimation accuracy are presented. An adaptive estimator which does not exploit any a priori information about the "true" system, is developed.