Mathematical Sciences: Estimation in Mixture Models

混合分布已被广泛用作模型，在这种情况下，数据被视为来自由若干亚种群以不同比例混合而成的种群。在这种情况下，模型识别可以被认为是对未知参数的估计，在最一般的情况下，这些参数将包括组件的数量、k、混合比例和来自组件分布的参数。申请拨款的目的是使首席研究员能够扩展其先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模型选择的现有研究。研究将集中在两个方面。首先，它将PI在单变量混合模型的最小海灵格距离估计上的工作扩展到多变量情况。这将涉及研究和推导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，与最大似然方法不同，它产生的估计对数据污染的存在既有效又稳健。其次，PI将考虑组件数量未知的情况。在这种情况下，可以获得其他参数的合理范围k的估计，然后使用模型选择标准从这些模型中进行选择。将考虑在这种情况下使用现有准则的调查，以及使用贝叶斯恢复最小化方法计算后验分布的新准则的应用。

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.