The Empirical Likelihood: an alternative for Signal Processing Estimation

This paper presents a new robust estimation scheme for signal processing problems. The empirical likelihood is a recent semi-parametric estimation method [1] which allows to estimate unknown parameters and to build confidence areas without using a prior model for the PDF: this method uses only information contained in the observed data when we have no prior distribution on the problem. However, in presence of priors on the parameter of interest, this information can be taken into account by means of constraints in an optimization problem. The aim of this paper is twofold: first, the empirical likelihood procedure is introduced in a very simple case and then, some priors on the unknown parameters are added in the study of a more elaborated problem. In order to illustrate this analysis, an example is studied all around this paper: the covariance matrix estimation from random data. In this particular case, a closed-form expression is derived for the solution of the corresponding optimization problem. Finally, theoretical results are emphasized by several simulations corresponding to real situations, which compare classical methods against the empirical likelihood method. Index Terms Empirical Likelihood, Maximum Likelihood, covariance matrix estimation, structured parameters estimation, statistical performance analysis, non-Gaussian noise. F. Pascal is with SATIE, ENS Cachan, CNRS, UniverSud, 61 Av. du Pdt Wilson, F-94235 Cachan Cedex, France (e-mail: [email protected]). H. Harari-Kermadec is with CREST-LS and University Paris-Dauphine, France (e-mail: [email protected]). P. Larzabal is with SATIE, ENS Cachan, CNRS, UniverSud, 61 Av. du Pdt Wilson, F-94235 Cachan Cedex, France (e-mail: [email protected]). October 9, 2007 DRAFT SUBMITTED TO IEEE TRANS. ON SIGNAL PROCESSING 2