Mathematical Sciences: Semiparametric Models and Empirical Processes

这项研究包含两个主要的研究主题，既有重叠的统计数据，也有概率。传统的依赖于参数模型的统计方法受限于参数在欧氏空间的一个子集，而非参数模型在数据高维时由于缺乏结构或约束而开始崩溃。本文将通过定义一类具有和切空间的模型来研究介于这两个极值之间的半参数模型。在这门课上，我们将讨论关于效率、假设检验、估计以及对各种截尾或截断数据的应用的统计问题。概率研究将集中在函数的Donsker类及其性质，椭球类的Donsker性质在基于经验度量的检验构造中的应用，以及自举经验过程的含义。

This research contains two major research themes, both overlapping statistics and probability. Traditional statistical methods relying on parametric models are limited by the restriction of parameters to a subset of Euclidean space, while nonparametric models begin to break down when the data is high- dimensional because of a lack of structure or restriction. This research will investigate semiparametric models which lie between the two extremes by defining a class of models with sum tangent spaces. For this class, statistical questions about efficiency, hypothesis testing, estimation and applications to various kinds of censored or truncated data will be addressed. Probability research will focus on Donsker classes of functions and their properties, the application of the Donsker property for ellipsoidal classes in the construction of tests based on empirical measures, and the implications of bootstrapping empirical processes.