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

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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.