  Multivariate normal mixtures provide a flexible model for high-dimensional
data. They are widely used in statistical genetics, statistical finance, and
other disciplines. Due to the unboundedness of the likelihood function,
classical likelihood-based methods, which may have nice practical properties,
are inconsistent. In this paper, we recommend a penalized likelihood method for
estimating the mixing distribution. We show that the maximum penalized
likelihood estimator is strongly consistent when the number of components has a
known upper bound. We also explore a convenient EM-algorithm for computing the
maximum penalized likelihood estimator. Extensive simulations are conducted to
explore the effectiveness and the practical limitations of both the new method
and the ratified maximum likelihood estimators. Guidelines are provided based
on the simulation results.
