Mathematical Sciences: Inference for Nonparametric Regresssion

研究了参数回归和非参数回归的推理问题。通过比较参数估计量和非参数估计量，提出了参数回归模型的拟合优度检验。这是完成相当一般的模型，包括广义线性和时间序列模型。利用中间渐近效率，提出了对这些非参数型检验进行解析性、渐近性比较的新方法。这些测试思想被扩展到更一般的拟合优度问题，例如测试可加性，其中零模型甚至是非参数的。本文还研究了其他推理问题，包括构造具有良好有限样本性质的非参数回归渐近有效置信区间的几个建议。通常有理由相信，在许多科学领域，如生物学、工程学、心理学等，所收集的数据是由两个组成部分产生的：一个是非随机组成部分，代表所有个体或主体共同的自然特征，另一个是随机组成部分，说明个体差异。非随机成分代表正在研究的现象的可再现或可预测的方面，因此具有相当大的兴趣。在许多情况下，从理论考虑或过去的经验，人们可以假设一个数学框架或模型，被认为是描述非随机成分。当这些模型正确时，它们可以提供有用的总结和预测工具。然而，不正确的模型可能导致误导性的结论和对未来事件的不准确预测。因此，重要的是要有方法来评估数据的假设模型的准确性。本研究的重点是为此目的开发新工具，这些工具依赖于对非随机成分的两种估计量的比较：即在假设模型正确的情况下计算的估计量和不采用该假设的非常灵活的估计量。在这项工作中解决的理论统计问题包括制定比较两个估计器的客观标准和确定这些标准的值，这些标准表明数据不支持假设模型。考虑的其他问题包括使用数据来构造具有已知的、指定的、包含数据的未知的、非随机成分的机会的区间。

DMS 96-25496 Eubank Problems of inference for parametric and nonparametric regression are investigated. Tests of goodness-of-fit for parametric regression models are proposed that derive from the comparison of parametric and nonparametric estimators. This is accomplished for rather general models that include generalized linear and time series models. New methods are developed for analytic, asymptotic comparison of these nonparametric type tests through the use of intermediate asymptotic efficiency. These testing ideas are extended to more general goodness-of-fit problems, such as testing for additivity, where the null model is even nonparametric. Other inference problems are also studied, including several proposals for the construction of asymptotically valid confidence intervals for nonparametric regression that have good finite sample properties. 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. The nonrandom component represents the reproducible or predictable aspect of the phenomenon being studied and is therefore of considerable interest. In many cases, from theoretical considerations or past experience, one can postulate a mathematical framework or model that is believed to describe the nonrandom component. When such models are correct they can provide useful summary and predictive tools. However, an incorrect model can lead to misleading conclusions and inaccurate predictions of future events. Thus, it is important to have methodology for assessing the accuracy of postulated models for data. This research focuses on the development of new tools for this purpose that rely on the comparison of two types of estimators for the nonrandom component: namely, an estimator that is computed under the assumpt ion that the postulated model is correct and a very flexible estimator that does not employ this assumption. The theoretical statistical issues that are addressed in this work include the development of objective criteria for comparing the two estimators and the determination of values for these criteria which indicate that the data does not support a postulated model. Other problems that are considered include the use of the data to construct intervals that have a known, specified, chance of containing the unknown, nonrandom component of the data.