Mathematical Sciences: Topics in Nonparametric Function Estimation

教授保罗L. Speckman将研究非参数曲线和曲面估计中的几个统计问题，即从噪声数据中恢复光滑曲线和曲面的问题。 曲线估计有很多常用的方法，包括核估计和光滑样条。 该项目旨在将这些方法应用于两个特定领域，构建曲线的置信集和估计平滑曲线中的变点（或断点）。 该项目的第一部分旨在为未知的光滑曲线开发可靠、实用的置信带。 研究人员将推导出必要的渐近理论，以基于交叉验证或其他数据驱动的平滑器构建置信带，并通过计算机模拟研究其有限样本特性。 第二部份将利用相关的分布理论，发展出侦测和估计变点以及估计变化幅度的方法。 在这种情况下，变点被定义为在其他平滑响应函数或其导数中发生跳跃不连续的点。 这些变点模型在各种应用中是有用的，包括在某些类型的临床试验中对患者的纵向研究。 教授保罗L. Speckman将调查几个统计问题，在一个地区被称为非参数曲线和曲面估计，恢复光滑曲线和曲面的问题，从嘈杂的数据。 这些问题出现在各种各样的科学领域，如经济学，大气科学，工程和医学。 目前有许多方法可用于估计这种曲线。 Speckman教授要解决的第一个问题是为这些估计提供现实和实用的误差范围。 这将使研究人员使用非参数曲线估计，以了解如何良好的， 结果是。 将通过计算机模拟获得理论结果并研究其适用性。 第二个要研究的问题，称为变点问题，涉及到在一个原本平稳的过程中突然发生变化的情况。一个例子是当输出质量突然变化时的工业过程。 其他例子来自对人类某些疾病的研究。 新方法将被测试的一个具体案例是某种形式的肌肉萎缩症，在疾病发作时肌肉力量突然开始下降。 在这个项目中开发的技术将使研究人员能够检测何时出现这种变化点，并估计变化的大小。

Professor Paul L. Speckman will investigate several statistical problems in nonparametric curve and surface estimation, the problem of recovering smooth curves and surfaces from noisy data. There are a number of general methods for curve estimation which have been extensively studied including kernel estimators and smoothing splines. This project is aimed at applications of these methods in two specific areas, constructing confidence sets for curves and estimating change-points (or breakpoints) in otherwise smooth curves. The first part of the project is aimed at developing reliable, practical confidence bands for unknown smooth curves. The investigator will derive the necessary asymptotic theory to construct confidence bands based on cross-validated or otherwise data-driven smoothers, and he will study their finite sample properties by computer simulation. The second part of the project will employ related distibutional theory to develop methods for detection and estimation of change-points and estimation of the magnitude of the change. In this context, change-points are defined to be points where jump discontinuities in an otherwise smooth response function or its derivatives occur. These change-point models are useful in a variety of applications including longitudinal studies of patients in certain kinds of clinical trials. Professor Paul L. Speckman will investigate several statistical problems in an area known as nonparametric curve and surface estimation, the problem of recovering smooth curves and surfaces from noisy data. Such problems arise in a wide variety of areas of science such as economics, atmospheric sciences, engineering and medicine. There are a number of methods currently available for estimating such curves. The first problem to be addressed by Professor Speckman will be that of giving realistic and practical error bounds for these estimates. This will enable investigators using nonparametric curve estimators to know how good their results are. Theoretical results will be obtained and their applicability will be studied through computer simulation. The second problem to be studied, known as a change-point problem, pertains to situations where there is a sudden change in an otherwise smooth process. One example is an industrial process when there is a sudden shift in the quality of the output. Other examples come from the study of certain diseases in humans. One specific case in which the new methods will be tested is a certain form of muscular dystrophy where muscle strength abruptly begins to decline at the onset of the disease. Techniques to be developed in this project will enable researchers to detect when such change-points have occurred and to estimate the size of the change.