Mathematical Sciences: Exploring Regression Structure Using Nonparamentric Functional Estimation

本研究主要研究外生变量数量较大时模型中外生变量的统计回归函数的形式选择问题。通常的做法是施加诸如线性或可加性之类的条件，以避免与高维稀疏数据相关的困难。平均导数函数的估计将用于选择或拒绝对回归形式的特定限制。所考虑的泛函是整型的，可以允许它们以通常的参数速率进行估计。本文将考虑基于核密度估计的这些泛函的估计量，并分析它们的大样本性质。这些估计量将被用来检验关于回归函数形式的各种假设。所提出的估计和测试的有限样本行为将使用构造数据和实际数据进行研究。在一定条件下，研究这些估计量在投影追踪回归中识别投影方向的应用。在投影追踪密度估计中，将使用与Fisher信息矩阵非常相似的函数来确定方向。

This research is devoted to the study of choice of the form of the statistical regression function in terms of the exogenous variables in the model when the number of exogenous variables is large. The common practice is to impose conditions such as linearity or additivity to avoid the difficulties associated with sparse data in high dimensions. Estimates of average derivative functionals will be used to select or reject a particular restriction on the form of the regression. The functionals considered are of integral type which may allow their estimation at the usual parametric rate. Estimators of these functionals which are based on kernel density estimators will be considered and their large sample properties analyzed. These estimators will then be used to test various hypotheses concerning the form of the regression function. The finite sample behavior of the proposed estimators and tests will be studied using constructed and real data. Under certain conditions these estimators will be studied for their use in identifying projection directions in projection pursuit regression. Similar functionals closely resembling the Fisher information matrix will be used to determine directions in projection pursuit density estimation.