Some Problems in Nonparametric Regression

非参数回归中的一些问题

• 批准号: 9970902
• 负责人: Randall Eubank
• 金额: $ 8.03万
• 依托单位: Texas A&M Research Foundation
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1999
• 资助国家: 美国
• 起止时间: 1999-07-15 至 2002-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9970902&HistoricalAwards=false
• 关键词: Some; Problems; Nonparametric; Regression

This project is concerned with a number of inference problems innonparametric regression analysis. Two of the problems being consideredinvolve the use of nonparametric smoothing methodology to assess thelack-of-fit of certain types of parametric models. The development of suchlack-of-fit tests has been an active research area for about the last 10years. However, effective methods for obtaining analytic assessments ofthe relative performance of such tests has not, as yet, been determined.One of the goals of this project is derivation of a framework for studyingthe relative asymptotic efficiency of nonparametric smoothing based testsusing an asymptotic intermediate efficiency approach that makes it possibleto extend the concept of asymptotic efficiency into the nonparametric, orinfinite dimensional, alternative setting. The other lack-of-fit testingproblem being studied concerns nonlinear parametric regression models.This problem is of particular interest since it provides a case where, insome instances, the usual smoothing parameter asymptotics obtain under thenull model and thereby allow for the development of asymptoticallydistribution free test statistics. Another collection of problems underconsideration is concerned with the derivation of suitable varianceestimators and associated heteroscedasticity diagnostics in the context ofpartially linear models. Variance estimators are needed here for a numberof reasons which include their use in testing hypotheses about theparametric components (e.g., treatment effects) of the model. A finalclass of problems under investigation concerns computational methods forboundary correcting smoothing spline estimators. These problems haveimplications and applications to the problem of interval estimation innonparametric regression that are also being explored.One of the most common approaches to statistical analysis involves thefitting of data by a parametric model. Such models are frequentlydeveloped through consideration of the physical nature of a problem understudy which may suggest a mathematical relationship or model for the data.For example, in Biology there are mathematical models that have beenproposed for relating growth (of humans, animals, etc.) to age, while inMeteorology there are mathematical models deriving from physics thatattempt to predict the development of storms and weather patterns. Thisproject is concerned, in part, with the study and development of variousstatistical methods for assessing the validity of parametric models. Themethods being considered use flexible data fitting techniques known asnonparametric smoothers to evaluate and compare fits obtained from aproposed or postulated parametric model. Those smoothes can be used toobtain statistical tests that can, in turn, be used to assess the accuracyof a model in question. This is particularly important because anincorrectly specified model can have potentially dangerous consequences inthat it can produce incorrect conclusions and predictions about the processunder study. In addition to the development of new testing methods,techniques are being developed for the comparison of different tests todetermine which type of test performs the best in different situations thatmight be encountered in practice. Other problems under study include thedevelopment of computationally efficient methods for computing certaintypes of data smoothers and methods for conducting statistical inferencewhen it may not be possible to completely specify a parametric model for aset of data. The latter problem arises frequently in practice where it maybe reasonable to assume a particular parametric form for a portion of themodel corresponding, for example, to presence or absence of cancer in asubject, but there is no obvious choice for a parametric model involvingother influential variables, such as the time of a subject's evaluations.

本课题研究非参数回归分析中的若干推论问题。 两个问题正在考虑involve使用非参数平滑方法来评估thack-of-fit的某些类型的参数模型。 最近10年来，这种拟合检验的发展一直是一个活跃的研究领域。 然而，有效的方法来获得分析评估的相对性能，这样的测试还没有，到目前为止，被determined.One的目标，这个项目是推导出一个框架，用于studyingthe相对渐近效率的非参数平滑为基础的testsusing渐近中间效率的方法，使之成为可能的渐近效率的概念扩展到非参数，或无限维，替代设置。 另一个正在研究的不适合检验问题涉及非线性参数回归模型，这个问题特别有趣，因为它提供了一种情况，在某些情况下，通常的平滑参数渐近性在thenull模型下获得，从而允许发展渐近分布自由检验统计量。 另一个正在考虑的问题是关于在部分线性模型的背景下，推导出合适的方差估计量和相关的异方差诊断。 这里需要方差估计量的原因有很多，包括它们在检验关于参数分量的假设时的用途（例如，治疗效果）。 最后一类问题的调查涉及边界校正光滑样条估计的计算方法。 这些问题对非参数回归中的区间估计问题也有影响和应用，这些问题也正在探索中。统计分析中最常见的方法之一是用参数模型拟合数据。 这类模型通常是通过考虑待研究问题的物理性质而建立起来的，这些物理性质可能暗示着数据的数学关系或模型。例如，在生物学中，有一些数学模型被提出来与（人类、动物等）的生长相联系。在气象学中，有一些数学模型是从物理学中推导出来的，它们试图预测风暴和天气模式的发展。 这个项目的部分内容是研究和发展各种统计方法来评估参数模型的有效性。 正在考虑的方法使用灵活的数据拟合技术（称为非参数平滑器）来评估和比较从提议或假设的参数模型中获得的拟合。 这些平滑可以用来获得统计测试，反过来，可以用来评估模型的准确性。 这一点特别重要，因为一个不正确的模型可能会产生潜在的危险后果，因为它可能会产生关于研究过程的错误结论和预测。 除了开发新的测试方法外，还开发了各种技术来比较不同的测试，以确定哪种类型的测试在实践中可能遇到的不同情况下表现最好。 正在研究的其他问题包括发展计算效率高的方法来计算某些类型的数据平滑和方法进行统计推断时，它可能不可能完全指定一个参数模型的一组数据. 后一个问题经常出现在实践中，它可能是合理的，假设一个特定的参数形式对应的一部分模型，例如，存在或不存在癌症的一个主题，但没有明显的选择参数模型involvingother有影响力的变量，如时间的主题的评价。