Collaborative Research: Estimation, Inference, and Computation for Finite Nonparametric Mixtures

本项目旨在发展估计非参数多元有限混合函数分量和权值的理论和方法。这些混合只是假设分量来自某一族多元密度函数，而没有任何参数说明。已经活跃在这一新兴研究领域的研究人员，将有限参数混合理论中已知的一些估计方法应用到非参数环境中。PI和co-PI提出了一些实际可行的快速算法，可以用来计算实际中得到的估计器。最后，两位研究者都展示了如何获得所提出的估计量的大样本渐近结果。分布的有限非参数混合可以为许多重要的实际问题提供答案。例如，在具有多个可能的诊断的复杂医疗条件的情况下，它们可以用来帮助医生建立明确的诊断。这种情况的一个例子是一名可能患有心脏病的患者，其他鉴别诊断也是可能的。发展心理学提供了另一个有用的例子。事实上，对儿童认知发展的研究，特别是对儿童完成各种任务所使用的策略的识别，也可以很容易地使用这些混合方法来建模。这对发展心理学具有重要的意义，为儿童心理学家面临的许多难题提供了答案，同时帮助儿童以最佳的方式成熟和发展。PI和co-PI提出了许多有效的算法来估计这些混合，并完成了上述实际任务。这些算法将作为名为MixTool的R软件包的一部分公开提供并易于使用。

This project aims to develop theory and methods for estimation of the functional components and weights in the nonparametric multivariate finite mixtures. These mixtures only assume that the components are drawn from some family of multivariate density functions without any parametric specification. The investigators, who are already active in this emerging area of research, adapt a number of estimation methods that are known from finite parametric mixture theory to the nonparametric context. The PI and the co-PI propose a number of practically feasible and fast algorithms that can be used to compute the resulting estimators in practice. Finally, both investigators show how to obtain large-sample asymptotic results for the proposed estimators. Finite nonparametric mixtures of distributions can provide answers to many practically important questions. As an example, they can be used to help a physician in establishing the definitive diagnosis in case of a complex medical condition with a number of possible diagnoses. An example of such a situation is a patient with a possible heart attack where other differential diagnoses are also possible. Developmental psychology provides another useful example. Indeed, study of cognitive development in children, in particular identification of strategies used by children to accomplish various tasks, can also be modeled easily using these mixtures. This has important implication for developmental psychology, providing answers to many difficult questions faced by child psychologists while helping children mature and develop in an optimal way. The PI and the co-PI propose a number of efficient algorithms to estimate these mixtures and accomplish the practical tasks mentioned above. These algorithms will be publicly available and easy to use as part of the R software package called mixtools.