Studies in Measurement Error Problems

该奖项是根据2009年美国复苏和再投资法案（公法111-5）资助的。首席研究员（P.I.）将研究测量误差模型中的三个主题，开发相关方法并分析其相应的特性和性能。第一个主题是关于主模型包含未指定误差的情况下的功能模型，因此它本身是一个半参数模型。P.I.将研究两个非参数成分，未知误差分布和未知潜变量分布的结构和相互作用，并提出一种最佳处理它们的操作。一般的方法是基于几何的。所得到的估计过程对两个组成部分的模型规格错误都具有鲁棒性，并且可以实现最优的估计效率。本文将从理论和数值实例两方面证明渐近一致性和正态性。第二个主题涉及测量误差模型中的模型拟合优度检验。私家侦探将提出一个伪分数类型的方法。她将证明新的测试程序在适应这些模型中特定的计算问题方面是可行的，并且具有所需的一致性和功率特性。此外，与通常的轮廓估计过程相关的最优功率特性可以通过投影等效地实现。她还将研究Wald测试和伪分数测试之间的关系，并在比以前已知的文献更广泛的范围内证明它们的等效性。第三个主题涉及测量误差模型中的小样本性能。已有文献表明，测量误差模型中的一阶渐近性往往需要非常大的样本量才能显示其相关性。P.I.将使用鞍点近似技术来解决这个问题，从而获得比经典一阶理论更高阶的近似。由于功能测量误差模型是半参数的，而现有的鞍点近似理论的发展严重依赖于参数模型假设，因此P.I.将在这一领域开发和研究新的方法。本提案中的一系列项目将以测量误差模型中最一般的形式解决一些最基本的问题。由于测量误差广泛存在于几乎所有科学领域，包括卫生和医学、环境和大气科学、金融和经济、材料和化学科学，新的方法将引起广泛的兴趣，并在这些领域具有重要的应用。它们还将激发统计科学本身在相关半参数问题和计算方法方面的进一步研究和发展。

This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5).The principal investigator (P.I.) will study three topics in measurement error models, develop relevant methodologies and analyze their corresponding properties and performances. The first topic concerns functional models in the situation when the main model contains unspecified error, hence is a semiparametric model by itself. The P.I. will study the structure and interaction of the two nonparametric components, the unknown error distribution and the unknown latent variable distribution, and propose an operation to best treat each of them. The general approach is geometry based. The resulting estimating procedure possesses robustness to model misspecification in both components, and allows to achieve optimal estimation efficiency. The asymptotic consistency and normality will be demonstrated both theoretically and in numerical examples.The second topic concerns the model goodness-of-fit test in measurement error models. The P.I. will propose a pseudo-score type methodology.She will demonstrate that the new testing procedure is feasible in accommodating the computational issues specific in such models, and has the desired consistency and power property. In addition, the optimal power property associated with the usual profiling estimation procedure can be equivalently achieved via projection. She will also study the relation between the Wald test and the pseudo-score test and demonstrate their equivalence in a wider range than previously known in literature.The third topic concerns the small sample performance in measurement error models. Existing literature has indicated that the first order asymptotics in measurement error models often require very large sample size to show its relevancy. The P.I. will tackle this problem using a saddle point approximation technique, hence achieving a higher order approximation than the classical first order theory. Because the functional measurement error model is semiparametric, yet existing saddle point approximation theory is developed and heavily relies on parametric model assumption, the P.I. will develop and study new methodology in this area.The series of projects in this proposal will resolve some of the most fundamental issues in their most general form in measurement error models. Since errors in measurements widely present in almost all scientific fields, including health and medicine, environment and atmospheric science, finance and economics, material and chemical sciences, the new methodologies will generate wide interest and have important application in these fields. They will also provoke further studies and development in related semiparametric problems and computing methods in statistical sciences itself.