Mathematical Sciences: Applications of Smoothing Splines forInference and Data Analysis

本研究的第一部分为样条估计函数和估计的平滑参数建立了有效的置信集。置信区间的两种方法是将估计的偏差视为随机变量或用上界代替偏差。第一种方法得到的是仅在平均意义上准确的短间隔，而第二种方法得到的是较长的保守间隔。平滑样条的置信区间由交叉验证函数的分布导出。本研究的第二部分将样条方法应用于显微镜和土壤科学中出现的两个不适定问题。这两个主题都与解决离散和噪声数据的积分方程有关。在这种情况下，部分样条模型可以补偿边界效应，而某些实验设计可以提高估计的分辨率。这个统计研究项目解决了在数据分析中估计平滑函数f的一般问题，从f的函数中观察到有误差。通常f的参数形式是未知的，因此考虑f的非参数估计是很重要的。本项目侧重于一个在各种应用中成功的非参数估计，即交叉验证的平滑样条。

The first part of this research develops valid confidence sets for functionals of a spline estimate and the estimated smoothing parameter. Two approaches to confidence intervals are to treat the bias of the estimate as a random variable or to replace the bias by an upper bound. The first method yields short intervals that are accurate only in an average sense while the second method gives longer conservative intervals. Confidence intervals for the smoothing spline are derived from the distribution of the cross-validation function. The second part of this research applies spline methods to two ill-posed problems that arise in microscopy and soil science. Both of these topcs are related to solving an integral equation with discrete and noisy data. In this context, partial spline models can compensate for boundary effects while certain experimental designs may imporve the resolution of the estimate. This statistical research project addresses the general problem in data analysis of estimating a smooth function f, from functionals of f that are observed with error. Often a parametric form of f is unknown and it is important to consider nonparametric estimators of f. This project focuses on one nonparametric estimator that has been successful over a diverse range of applications, a cross-validated smoothing spline.