Theory and applications of finite mixture models

数据的异质性是常态，而同质性是例外。混合模型提供了处理数据异构性的灵活方法。然而，从经典参数似然推断的角度来看，混合模型的统计推断由于问题的不规则性而特别具有挑战性。因此，混合模型通常被视为非正则统计模型中最重要的一类。该研究项目的长期目标是在混合模型、经验似然及其接口领域开发创新方法，并使用这些方法解决各种应用中没有简单解决方案的统计问题。本提案由三部分组成。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。这个研究项目是新颖的，多方面的。它将利用几个重要的统计工具，如混合模型，功能数据分析，经验似然方法来解决重要的科学问题。提出了若干值得关注的理论问题，此外，本研究项目的成功将对混合模型的理论和应用产生重大贡献。反过来，这些结果将有助于解决自然科学和工程中许多领域出现的问题。所开发的理论和方法也将有很大的扩展和应用范围。

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.