Mathematical Sciences: Estimation in Mixture Models

混合分布已被广泛用作模型的情况下，数据被视为来自一个人口，是由几个亚人口以不同的比例混合。 在这种情况下，模型识别可以被认为是获得估计的未知参数，其中在最一般的情况下，将包括组件的数量，$k$，混合比例和参数的组件分布。 拟议补助金的目的是使首席研究员能够扩展她以前在混合模型最小距离估计方面的工作，并更加熟悉混合模型背景下的模型选择方面的现有研究。 研究将集中在两个领域。首先，它将扩展PI的工作，最小Hellinger距离估计单变量混合模型的多变量的情况下。这将涉及调查和推导出一个适当的多元密度估计混合模型，以及考虑它的计算方面，这种方法的优点是，它产生的估计，是有效的和强大的数据污染的存在不同的最大似然方法。 其次，PI将考虑组件数量未知的情况。 在这种情况下，人们可以获得其他参数的合理范围内的$k$的估计，然后从这些模型中选择使用模型选择标准。 调查使用现有的标准在这种情况下，以及使用贝叶斯恢复最小化方法计算后验分布的一个新的标准的应用程序将被考虑。

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.