Developments and challenges in mixture models

混合模型提供了非常灵活和有效的方法来处理数据中的异构性。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。尽管混合模型的应用非常广泛，但对混合模型的统计理论和方法的研究仍然有限。本研究计划旨在发展混合模型领域的创新方法，并利用这些方法解决存在异质性的统计问题。短期目标是研究对一般位置-尺度混合顺序的推断；发展伽玛分布有限混合的理论和方法；探索混合模型在DNA甲基化研究中的应用。提出的研究方案将极大地促进当前混合模型的研究，特别是对有限混合模型各方面的基本推理问题的研究。提议的研究是由我正在进行的和未来的合作所激发的，这些合作涉及遗传学、生物学、工程学、医学和金融方面的问题。潜在的研究成果将为异构数据的分析提供新的见解和有用的指导。

Mixture models provide extremely flexible and effective 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. Despite the extremely wide applications of mixture models, research in statistical theory and methods on mixture models is still limited. The proposed research program aims to develop innovative methodologies in the field of mixture models and to use such methodologies in solving statistical problems in the presence of heterogeneity. The short-term objectives are to investigate inference on the order of general location-scale mixtures; to develop theory and methodologies for finite mixtures of gamma distributions; and to explore the use of mixture models in DNA methylation studies. The proposed research program will significantly enhance the current research on mixture models, especially the fundamental inference problems on various aspects of finite mixture models. The proposed research is motivated by my ongoing and future collaborations arising from problems in genetics, biology, engineering, medicine, and finance. The potential research outcomes will provide new insights and useful guidance for the analysis of heterogeneous data.