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Maximum likelihood estimation of Gaussian mixture models with differentclass-specific covariance matrices is known to be problematic. This is due tothe unboundedness of the likelihood, together with the presence of spuriousmaximizers. Existing methods to bypass this obstacle are based on the fact thatunboundedness is avoided if the eigenvalues of the covariance matrices arebounded away from zero. This can be done imposing some constraints on thecovariance matrices, i.e. by incorporating a priori information on thecovariance structure of the mixture components. The present work introduces aconstrained equivariant approach, where the class conditional covariancematrices are shrunk towards a pre-specified matrix Psi. Data-driven choices ofthe matrix Psi, when a priori information is not available, and the optimalamount of shrinkage are investigated. The effectiveness of the proposal isevaluated on the basis of a simulation study and an empirical example.

A data driven equivariant approach to constrained Gaussian mixture modeling