Developments and challenges in mixture models

混合模型在数据的统计分析中具有相当大的价值，因为它们提供了一种非常灵活的建模方式。同时，由于传统的参数推理理论往往不能直接应用于混合模型中的问题，因此混合模型的研究尤其具有挑战性。该提案的目的是在混合模型和经验似然方法领域发展理论和创新方法，以处理各种应用中没有简单解决方案的统计问题。本提案由三个项目组成。在第一个项目中，将进行具有标量响应和功能预测因子的回归混合物的新模型。将使用功能主成分技术实现降维。然后将讨论由此产生的回归混合物中的顺序选择。混合模型在遗传学中有许多自然的应用。在第二个项目中，混合模型将用于检测存在位点和/或等位基因异质性的等位基因关联。由于混合模型方法考虑了基因座异质性，因此在关联分析方面比其他现有方法具有更大的功效。经验似然是一种基于数据驱动的似然比函数的非参数推理方法，在过去的十五年中已经成为一种广泛使用和研究的统计方法。在第三个项目中，将探索经验似然方法，利用辅助信息在缺失数据问题中产生有效的推理。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、功能数据分析和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。

Mixture models are of considerable value in the statistical analysis of data because they provide a very flexible way of modeling. At the same time, the subject of mixture models is particularly challenging because the traditional theory of parametric inference often can not be applied directly to problems in mixture models. The objective of the proposal is to develop theory and innovative methodology in the field of mixture models and empirical likelihood methods to cope with statistical problems in a variety of applications for which no simple solution exists. This proposal is organized by three projects. In the first project, a novel modeling of regression mixtures with scalar responses and functional predictors will be carried out. The functional principal component techniques will be used to achieve dimension reduction. Order selection in the resulting regression mixture will then be addressed. Mixture models have many natural applications in genetics. In the second project, mixture models will be used to detect allelic association in the presence of locus and /or allelic heterogeneity. Since the mixture model approach takes into account locus heterogeneity, it is expected to have greater power in association analysis than other existing methods. The empirical likelihood, which is a nonparametric method of inference based on a data-driven likelihood ratio function, has become a widely used and studied statistical method over the past fifteen years. In the third project, the empirical likelihood method will be explored to produce efficient inference in missing data problems by using auxiliary information. This research program is novel and multi-faceted. It will draw on several important statistical tools such as mixture models, functional data analysis, and 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 application 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.