Mathematical Sciences: Exploring Regression Structure Using Nonparamentric Functional Estimation

本研究致力于研究在外生变量较多的情况下，模型中外生变量对统计回归函数形式的选择。通常的做法是强加一些条件，如线性或可加性，以避免与高维中的稀疏数据相关的困难。平均导数泛函的估计将被用来选择或拒绝对回归形式的特定限制。所考虑的泛函是积分型的，可以用通常的参数速率进行估计。我们将考虑这些基于核密度估计的泛函的估计量，并分析它们的大样本性质。然后，这些估计器将被用来检验关于回归函数形式的各种假设。我们将使用构造的和真实的数据来研究所建议的估计器和检验的有限样本行为。在一定条件下，这些估计量将被用于识别投影寻踪回归中的投影方向。在投影寻踪密度估计中，将使用与Fisher信息矩阵非常相似的相似泛函来确定方向。

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.