Maximum likelihood estimation of heterogeneous mixtures of Gaussian and uniform distributions

Existence and consistency of the Maximum Likelihood estimator of the parameters of heterogeneous mixtures of Gaussian and uniform distributions with known number of components are shown under constraints to prevent the likelihood from degeneration and to ensure identifiability. The EM-algorithm is discussed, and for the special case with a single uniform component a practical scheme to find a good local optimum is proposed. The method is compared theoretically and empirically to the estimation of a Gaussian mixture with “noise component” as introduced by Banfield and Raftery (1993) to find out whether it is a worthwhile alternative particularly in situations with outliers and points not belonging to the Gaussian components.