Some Problems in Nonparametric Regression

本课题主要研究非参数回归分析中的若干推断问题。正在考虑的两个问题涉及使用非参数平滑方法来评估某些类型的参数模型的拟合不足。在过去的10年里，这类不匹配测验的发展一直是一个活跃的研究领域。然而，获得这类测试的相对性能的分析评估的有效方法尚未被确定。本项目的目标之一是推导出一个框架，用于研究基于非参数平滑的测试的相对渐近效率，该方法使用渐近中间效率方法，使得将渐近效率的概念扩展到非参数、或无限维的替代设置成为可能。另一个正在研究的失配检验问题是关于非线性参数回归模型的，这个问题特别令人感兴趣，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是在零模型下获得的，从而允许发展渐近分布的无检验统计量。正在考虑的另一组问题涉及在部分线性模型的背景下导出适当的方差估计器和相关的异方差诊断。这里需要方差估计器的原因有很多，包括它们在检验关于模型的参数成分(例如，治疗效果)的假设时的使用。正在研究的最后一类问题涉及边界修正平滑样条估计器的计算方法。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，这些问题也在探索中。统计分析的最常见方法之一是用参数模型来拟合数据。这样的模型通常是通过考虑待研究问题的物理性质而开发的，这可能为数据提出了一种数学关系或模型。例如，在生物学中，有人提出了与(人、动物等)生长相关的数学模型。随着年龄的增长，在气象学中，有一些数学模型源自物理学，试图预测风暴和天气模式的发展。该项目在一定程度上涉及评估参数模型有效性的各种统计方法的研究和发展。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，来评估和比较从提出的或假设的参数模型获得的拟合。这些平滑可以用来获得统计检验，而这些检验又可以用来评估所讨论的模型的准确性。这一点特别重要，因为一个不正确指定的模型可能会产生潜在的危险后果，因为它可能会对正在研究的过程产生错误的结论和预测。除了开发新的测试方法外，正在开发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开发计算效率高的方法来计算某些类型的数据平滑器，以及在可能不可能为一组数据完全指定参数模型时进行统计推断的方法。后一个问题在实践中经常出现，其中假设模型的一部分对应于例如对象中癌症的存在或不存在的特定参数形式是合理的，但对于涉及其他影响变量的参数模型没有明显的选择，例如对象的评估时间。

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.