Theory and applications of finite mixture models

数据的异构性是常态，而同质性是例外。混合模型提供了灵活的方法来处理数据中的异构性。然而，从经典的参数似然推断的角度来看，混合模型的统计推断尤其具有挑战性，因为问题的不规则性。因此，混合模型通常被认为是非正则统计模型中最重要的一类。这一研究计划的长期目标是在混合模型、经验似然及其界面领域开发创新的方法，并将这些方法用于解决各种应用中的统计问题，这些应用没有简单的解决方案。这项建议由三个部分组成。短期目标如下：1)发展半参数函数混合回归模型的理论和方法；2)探索混合模型在循环数据分析中的应用；3)探索经验似然方法在多元混合模型中的应用。这一研究项目新颖、多方位。它将利用几个重要的统计工具，如混合模型、函数数据分析、经验似然方法来解决重要的科学问题。它提出了几个令人感兴趣的理论问题，此外，该研究项目的成功将对混合模型的理论和应用做出重大贡献。反过来，这些结果将对自然科学和工程中许多领域出现的问题有用。所发展的理论和方法也将有很大的推广和应用空间。

Heterogeneity in the data is the norm, while homogeneity is the exception. Mixture models provide flexible means of handling heterogeneity in data. However, statistical inference for mixture models is particularly challenging due to the irregularity of the problem from a classical parametric likelihood inference point of view. Therefore, mixture models are often viewed as one of the most important classes of non-regular statistical models. The long-term goal of this research program is to develop innovative methodologies in the fields of mixture models, empirical likelihood and their interface, and to use such methodologies in solving statistical problems in a variety of applications for which no simple solutions exist. This proposal is composed of three parts. The short term objectives are as follows: 1) to develop theory and methodology for semiparametric functional mixture regression models; 2) to explore the use of mixture models in circular data analysis; 3) to explore the use of empirical likelihood methods in multivariate mixture models. This research program is novel and multi-faceted. It will draw on several important statistical tools such as mixture models, functional data analysis, empirical likelihood methods to tackle important scientific questions. It raises several theoretical problems of interest, and in addition, success of this research program will yield significant contribution to the theory and applications of mixture models. In turn, these results will be useful in problems arising in many fields in the natural sciences and engineering. The theory and methodologies developed will also have great scope for extensions and applications.