Some Problems in Nonparametric Regression

本课题研究非参数回归分析中的一些推理问题。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，这种不匹配测试的发展一直是一个活跃的研究领域。然而，对这些测试的相对性能进行分析评估的有效方法尚未确定。该项目的目标之一是推导一个框架，用于研究基于非参数平滑的测试的相对渐近效率，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中成为可能。另一个正在研究的缺乏拟合检验问题涉及非线性参数回归模型。这个问题是特别有趣的，因为它提供了一种情况，在某些情况下，通常的平滑参数在全模型下获得渐近分布，从而允许发展无渐近分布的检验统计量。另一组正在考虑的问题是在部分线性模型的背景下推导合适的方差估计量和相关的异方差诊断。这里需要方差估计器的原因有很多，其中包括它们在检验关于模型的参数成分（例如，处理效果）的假设中的使用。研究的最后一类问题涉及边界校正平滑样条估计的计算方法。这些问题对非参数回归的区间估计问题也有启示和应用。统计分析最常用的方法之一是用参数模型拟合数据。这种模型通常是通过考虑问题的物理性质来开发的，这可能暗示了数据的数学关系或模型。例如，在生物学中，有人提出了将（人类、动物等）生长与年龄联系起来的数学模型，而在气象学中，有从物理学中衍生出来的数学模型，试图预测风暴和天气模式的发展。本项目部分涉及研究和开发用于评估参数模型有效性的各种统计方法。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测试，而统计测试反过来又可以用来评估模型的准确性。这一点尤其重要，因为一个不正确指定的模型可能会产生潜在的危险后果，因为它可能会对所研究的过程产生不正确的结论和预测。除了开发新的测试方法外，还在开发用于比较不同测试的技术，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某些类型数据平滑的有效方法，以及在不可能完全指定数据集的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，在这种情况下，假设模型的一部分对应于特定的参数形式可能是合理的，例如，受试者中是否存在癌症，但是对于涉及其他影响变量的参数模型来说，没有明显的选择，例如受试者评估的时间。

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.