Maximum likelihood estimation in constrained parameter spaces for mixtures of factor analyzers

Mixtures of factor analyzers are becoming more and more popular in the area of model based clustering of high-dimensional data. According to the likelihood approach in data modeling, it is well known that the unconstrained log-likelihood function may present spurious maxima and singularities and this is due to specific patterns of the estimated covariance structure, when their determinant approaches 0. To reduce such drawbacks, in this paper we introduce a procedure for the parameter estimation of mixtures of factor analyzers, which maximizes the likelihood function in a constrained parameter space. We then analyze and measure its performance, compared to the usual non-constrained approach, via some simulations and applications to real data sets.