Mathematical Sciences: Asymptotically Efficient Semiparametric Estimation

该方法主要针对无限维参数的估计问题，如累积分布函数、特征函数、回归模型中的误差分布等。在方法上，已成功地发展了一种几何方法来解决I.I.D.中的非参数估计问题。(独立且同分布)数据。这种方法也适用于广泛的非参数和半参数问题，其中传统的限制性I.I.D.假设是可以放松的。主要的重点是可微统计泛函的估计。对于未知抽象参数的一般可微变换，将利用几何方法构造渐近有效估计。在医学、计量经济学和工程可靠性研究等领域收集的复杂统计数据通常无法使用传统统计方法进行分析。临床试验人员的多样性，影响经济产出的多种投入，以及影响尖端电子或设备性能的部件之间错综复杂的相互作用，都不允许使用简单的统计建模技术。相反，这些应用程序需要一种更灵活且不会不必要地牺牲信息的新方法。拟议的研究将开发评估复杂统计数据的关键特征的方法。这些方法不依赖于传统假设的有效性。这项工作的结果将适用于各种各样的领域，包括药物疗效的估计、潜在患者的人口概况、工业产出的熟练程度特征以及电子设备成功操作的可能性。

The proposal is focused on estimation problems involving infinite dimensional parameters, such as cumulative distribution functions, characteristic functions, error distributions in regression models etc. Methodologically, a geometric approach has been successfully developed to solve nonparametric estimations problems from i.i.d. (independent and identically distributed) data. This approach is also applicable to a wide range of nonparametric and semiparametric problems, in which the traditional restrictive i.i.d. assumptions can be relaxed. The main emphasis is on the estimation of differentiable statistical functionals. For a general differentiable transformation of the unknown abstract parameter, the asymptotically efficient estimates will be constructed using the geometric approach. Complex statistical data that are collected in areas such as medical, econometric, and engineering reliability studies often cannot be analyzed using traditional statistical methods. The diversity of people in clinical trials, the multitude of inputs affecting economic outputs, and the intricate interaction of components affecting the performance of sophisticated electronics or equipment do not permit the use of simple statistical modeling techniques. These applications require instead a new methodology that is more flexible and yet does not sacrifice information unnecessarily. The proposed research will develop methods for estimating key features of complex statistical data. These methods do not depend on the validity of traditional assumptions. Results of this work will be applicable to a rich variety of areas, including the estimation of drug efficacy, a population profile of potential patients, proficiency characteristics for industrial output, and the probability of successful operation of electronic equipment.