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

在数据独立同分布的半参数模型中，建立了有效估计感兴趣参数的一般抽象理论。这个项目将从两个方面扩展这一理论。首先是建立有效影响函数的令人满意的演算。二是对相关数据理论的推广。该研究将为构建大量统计模型的估计量提供统一的观点和严格的数学基础。当模型仅部分地指定了决定特征或属性时，新观测的预期行为仍然可以使用所谓的半参数或非参数模型进行建模。这项工作做得有多好，取决于数据中的信息被利用得有多充分，以及在给定数据的情况下如何精确地表达模型。这项研究的重点是抽象必要的数学性质，为未来的观察提供良好的估计和预测。然后可以推导出这些期望性质的数学要求。

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.