A data driven equivariant approach to constrained Gaussian mixture modeling

Maximum likelihood estimation of Gaussian mixture models with different class-specific covariance matrices is known to be problematic. This is due to the unboundedness of the likelihood, together with the presence of spurious maximizers. Existing methods to bypass this obstacle are based on the fact that unboundedness is avoided if the eigenvalues of the covariance matrices are bounded away from zero. This can be done imposing some constraints on the covariance matrices, i.e. by incorporating a priori information on the covariance structure of the mixture components. The present work introduces a constrained approach, where the class conditional covariance matrices are shrunk towards a pre-specified target matrix $$\varvec{\varPsi }.$$ Ψ . Data-driven choices of the matrix $$\varvec{\varPsi },$$ Ψ , when a priori information is not available, and the optimal amount of shrinkage are investigated. Then, constraints based on a data-driven $$\varvec{\varPsi }$$ Ψ are shown to be equivariant with respect to linear affine transformations, provided that the method used to select the target matrix be also equivariant. The effectiveness of the proposal is evaluated on the basis of a simulation study and an empirical example.