Mathematical Sciences: Inference for Nonparametric Function Estimators

在这个项目的过程中，将研究非参数函数估计中的一些推理问题。要考虑的问题包括i)回归函数或其导数的断点的估计，ii)开发用于非参数回归的实用置信带，iii)在各种设置下使用非参数函数估计方法构建诊断拟合度检验，包括删失数据和加性模型，以及iv)具有相关误差的样条平滑。在许多科学领域，如生物学、工程学、心理学等，人们通常有理由相信，正在收集的数据是由两个组成部分产生的：一个非随机成分，代表所有个人或受试者共同的自然特征；以及一个随机成分，说明个体差异。这种非随机成分代表了正在研究的现象的可理解或可预测的方面，因此引起了相当大的兴趣。非参数函数估计是一个统计学领域，涉及发展灵活的方法来估计或描述数据的非随机成分。目前的研究项目涉及非参数函数估计中的各种问题，包括i)开发用于估计在某些现象中发生突变的点的程序(例如，定位河流的排水量或商品价格的突然变化点)ii)构建精度度量(类似于与民调相关的度量，等等)。伴随着非参数函数估计器，以及iii)统计方法的发展，以评估科学家假设的描述他们正在研究的各种过程的模型的准确性。

A number of inference problems in nonparametric function estimation will be investigated during the course of this project. The problems to be considered include i) estimation of breakpoints for regression functions or their derivatives, ii) development of practical confidence bands for use in nonparametric regression, iii) construction of diagnostic goodness-of-fit tests using nonparametric function estimation methodology in a variety of settings, including censored data and additive models, and iv) spline smoothing with correlated errors. It is often reasonable to believe in many areas of science, such as biology, engineering, psychology, etc., that the data being collected is produced by two components: a nonrandom component, representing a characteristic of nature common to all individuals or subjects, and a random component that accounts for individual variation. This nonrandom component represents the understandable or predictable aspect of the phenomenon being studied and is therefore of considerable interest. Nonparametric function estimation is an area of statistics concerned with the development of flexible methods for estimating or describing the nonrandom components of data. The current research project deals with a variety of problems in nonparametric function estimation including i) the development of procedures for estimating a point where an abrupt change has occurred in some phenomenon (e.g., locating a point of sudden change in the discharge of water from a river or prices of a commodity) ii) the construction of accuracy measures (similar to those associated with popularity polls, etc.) to accompany nonparametric function estimators and iii) the development of statistical methods to assess the accuracy of models that have been postulated by scientists to describe various processes they are studying.