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The robust improper maximum likelihood estimator (RIMLE) is a new method forrobust multivariate clustering finding approximately Gaussian clusters. Itmaximizes a pseudo-likelihood defined by adding a component with improperconstant density for accommodating outliers to a Gaussian mixture. A specialcase of the RIMLE is MLE for multivariate finite Gaussian mixture models. Inthis paper we treat existence, consistency, and breakdown theory for the RIMLEcomprehensively. RIMLE's existence is proved under non-smooth covariance matrixconstraints. It is shown that these can be implemented via a computationallyfeasible Expectation-Conditional Maximization algorithm.

Consistency, breakdown robustness, and algorithms for robust improper maximum likelihood clustering