Mathematical Sciences: Estimation in Mixture Models

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Mixtures distributions have been used extensively as models in situations where data are viewed as arising from a population that is comprised of several subpopulations mixed in varying proportions. In this context, model identification can be thought of as obtaining estimates for the unknown parameters which in the most general situation will include the number of components, $k$, the mixing proportions and the parameters from the component distributions. The purpose of the proposed grant is to enable the Principal Investigator to extend her previous work on minimum distance estimation for mixture models and to become more familiar with the existing research on model selection in the context of mixture models. The research will focus on two areas. First, it will extend the work of the PI on minimum Hellinger distance estimation for univariate mixture models to the multivariate case. This will involve investigating and deriving an appropriate multivariate density estimator for mixture models as well as considering the computational aspects of it. The advantage of this method is that it produces estimates that are both efficient and robust to the presence of data contamination unlike maximum likelihood methods. Second, the PI will consider the case when the number of components is unknown. In this case one can obtain estimates for the other parameters for a reasonable range of $k$'s and then choose from among those models using a model selection criterion. Investigation of the use of existing criteria in this situation as well as application of a new criterion using Bayesian Restoration Minimization method for computing posterior distributions will be considered.