  An important and yet difficult problem in fitting multivariate mixture models
is determining the mixture complexity. We develop theory and a unified
framework for finding the nonparametric maximum likelihood estimator of a
multivariate mixing distribution and consequently estimating the mixture
complexity. Multivariate mixtures provide a flexible approach to fitting
high-dimensional data while offering data reduction through the number,
location and shape of the component densities. The central principle of our
method is to cast the mixture maximization problem in the concave optimization
framework with finitely many linear inequality constraints and turn it into an
unconstrained problem using a "penalty function". We establish the existence of
parameter estimators and prove the convergence properties of the proposed
algorithms. The role of a "sieve parameter'' in reducing the dimensionality of
mixture models is demonstrated. We derive analytical machinery for building a
collection of semiparametric mixture models, including the multivariate case,
via the sieve parameter. The performance of the methods are shown with
applications to several data sets including the cdc15 cell-cycle yeast
microarray data.
