Mathematical Sciences: Semiparametric Models and Empirical Processes

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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.