Mathematical Sciences: Some Problems in Nonparametric Regression

Eubank提出研究非参数回归中的一些问题。其中最有趣的是鲁棒样条平滑问题。回归函数的典型样条估计量是通过最小化误差平方和加上惩罚函数得到的。然而，误差平方和对异常值非常敏感。通过用绝对误差代替平方误差（或误差的另一个适当函数），可以使估计器对异常值更健壮。尤班克斯将研究在这些更合适和更一般的误差函数中纳入和估计误差项的比例因子的影响。他还将研究潜在回归函数的折刀置信区间。这些非参数回归模型也将应用于传统的方差分析、协方差分析和类内回归模型。他将解决的另一个问题是在分位数函数及其密度估计中结合有序统计量的协方差结构。非参数回归解决了个体观察对的两个变量（如身高和体重）的关联问题。如果某些信息被错误编码，或者数据中包含了极不寻常的个体，那么找到两个变量之间的函数关系就会变得相当困难。尤班克将研究寻找功能关系的方法，这种方法对样本中的异常值不那么敏感，更能代表整个人口。

Eubank proposes to examine a number of problems in nonparametric regression. One of the most interesting is the problem of robust spline smoothing. The typical spline estimator of a regression function is found by minimizing a sum of squared errors plus a penalty function. However the sum of squared errors is highly sensitive to outliers. By substituting absolute error for squared error (or another appropriate function of the error) one can make the estimator more robust to outliers. Eubanks will examine the effect of incorporating and estimating a scale factor for the error terms in these more appropriate and general functions of the errors. He will also examine jackknife confidence intervals for the underlying regression function. These nonparametric regression models will also be applied to the traditional analysis of variance, analysis of covariance and intra-class regression models. Another problem he will address is the incorporation of the covariance structure of order statistics in the estimation of quantile functions and their densities. The problem of relating two variables (such as height and weight) for pairs of observations on individuals is addressed by nonparametric regression. Finding a functional relationship between the two variables is made considerably harder if some information is miscoded or extremely unusual individuals are included in the data. Eubank will investigate methods of finding functional relationships that will be less sensitive to outliers in the sample and more representative of the whole population.