Caractérisation des performances minimales d’estimation pour des modèles d’observation non-standards

Chengfang Ren
Thesis defended on September 28, 2015, 2:00 PM at CentraleSupelec (Gif-sur-Yvette) Amphi Janet

In the parametric estimation context, estimators performances can be characterized, inter alia, by the mean square error and the resolution limit. The first quantifies the accuracy of estimated values and the second defines the ability of the estimator to allow a correct resolvability. This thesis deals first with the prediction the "optimal" MSE by using lower bounds in the hybrid estimation context (i.e. when the parameter vector contains both random and non-random parameters), second with the extension of Cramér-Rao bounds for non-standard estimation problems and finally to the characterization of estimators resolution. This thesis is then divided into three parts:

- First, we fill some lacks of hybrid lower bound on the MSE by using two existing Bayesian lower bounds: the Weiss-Weinstein bound and a particular form of Ziv-Zakai family lower bounds. We show that these extended lower bounds are tighter than the existing hybrid lower bounds in order to predict the optimal MSE.

- Second, we extend Cramer-Rao lower bounds for uncommon estimation contexts. Precisely: (i) Where the non-random parameters are subject to equality constraints (linear or nonlinear). (ii) For discrete-time filtering problems when the evolution of states is defined by a Markov chain. (iii) When the observation model differs to the real data distribution.

- Finally, we study the resolution of the estimators when their probability distributions are known. This approach is an extension of the work of Oh and Kashyap and the work of Clark to multi-dimensional parameters estimation problems.


Jury members :
M. Jean-Yves Tourneret  Professeur à l’INP-ENSEEIHT Toulouse  (Rapporteur)
M. Philippe Forster  Professeur à l’Université Paris-Ouest  (Rapporteur)
M. Cédric Richard  Professeur à l’Université Nice Sophia-Antipolis  (Examinateur)
M. Karim Abed-Meraim  Professeur à l’Université d’Orléans  (Examinateur)
M. Éric Chaumette  Professeur à l’ISAE  (Encadrant de thèse)
M. Jérôme Galy   Maître de conférences à l’Université de Montpellier  (Encadrant de thèse)
M. Alexandre Renaux  Maître de conférences à l’Université Paris-Sud  (Directeur de thèse)

Keywords: Parametric estimation, maximum likelihood estimator, maximum a posteriori estimator, performance analysis, hybrid estimation, lower bounds on the mean square error, statistical resolution limit.