On optimal estimations with minimum error entropy criterion

This paper investigates the robustness, uniqueness, sufficient condition and the necessary condition for the minimum error entropy (MEE) estimation. For the robustness aspect, we show that the MEE estimator for a Gaussian nominal model is robust with respect to a relative entropy mismodeling criterion as well as the minimum mean-square error (MMSE) estimations. For the uniqueness aspect, we demonstrate by means of examples that for the singular case, the optimum solution of the MEE estimation will be nonunique. For the sufficient and the necessary condition, the former is established by the independence condition, and the later by score orthogonality condition. A specific example illustrates that the score orthogonality condition is just a necessary condition and not a sufficient one, because if an estimator satisfies the score orthogonality condition, it may be a local minimum or even a local maximum of the error entropy in a certain direction.