Mathematical Sciences: On the Construction of Efficient Estimates in Semi-Parametric and Nonparametric Models

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A general abstract theory has been developed for the construction of efficient estimates of the parameter of interest in semiparametric models when the data are independently and identically distributed. This project will expand this theory in two directions. First is the development of a satisfactory calculus for the construction of the efficient influence function. Second is the extension of the theory for dependent data. This research will provide a unified view and a rigorous mathematical basis for constructing estimators for a large class of statistical models. When models are only partially specified in terms of the determining features or attributes, the anticipated behavior for new observations can be modeled nonetheless using so-called semiparametric or nonparametric models. How well this is done depends on how fully the information in the data is utilized and on how precisely the model can be expressed given the data. This research focuses on abstracting the necessary mathematical properties for good estimates and good predictions for future observations. Then the mathematical requirements for these desired properties can be derived.