Abstract : This thesis deals with the study of estimators performance in the statistical signal processing framework and, more particularly, with the minimal bounds on the mean square error (MSE).
The main difficulty comes from the fact that the MSE of an estimator of a parameter with a finite support exhibits three areas : the asymptotic area, characterized by a large number of observations or a low noise level, where the estimation error is low, the threshold area where the MSE increases dramatically and the a priori area where the noise is high and where the observations do not bring informations. Many results concerning the asymptotic area are already available: distribution of the estimates, bias and variance. On the other hand, the non-asymptotic areas have been less studied.
The goal of this thesis is twofold. First, to pursue the investigation in the asymptotic area (particularly in terms of high signal to noise ratio with a finite number of observations) of direction of arrivals maximum likelihood estimators. Secondly, to give methods in order to predict the fundamental limits of an estimator on the three areas. The tool used here will be the minimal bounds on the MSE other than the Cramér-Rao bound for which the validity only holds in the asymptotic area. A main contribution is to propose a unified framework for the Bayesian bounds.
The obtained results have been applied in the spectral analysis context and in the carrier estimation context. They bring an interesting tool in order to predict the threshold phenomena.