Semiparametric Models for Correlated Data: The Quadratic Inference Function Approach

本研究的重点是一种新的统计方法来分析相关数据，二次推理函数（QIF）方法（Qu, Lindsay & Li 2000）。QIF建立在由一组平均零估计函数定义的半参数框架上，但与标准估计函数方法不同的是，它的方程多于未知参数的数量。与估计函数方法相比，QIF具有优点，例如不需要指定似然函数。它还克服了估计函数方法的局限性，如缺乏用于测试的目标函数和似然函数。提出的项目的主要目标之一是探索在平均零假设不满足时，相对于估计量的一致性的稳健性的QIF。第二个目标侧重于丢失数据问题，这在纵向数据中经常发生。一般来说，测试丢失的数据是否可以忽略仍然是一个具有挑战性的问题。QIF的拟合优度检验似乎是对不可忽略的缺失数据的有效检验。第三个目标是使用QIF测试相关数据的顺序限制替代假设。目前现有的测试工具并不令人满意，主要基于参数模型的似然函数，因此不适用于似然函数难以表述的相关数据。QIF与经验似然有关（Owen, 1988），这在非参数模型中很流行。提出的项目还说明了QIF的Edgeworth扩展，并探讨了如何应用自举策略来提高小样本相关数据的测试精度。本研究将在生物统计学、计量经济学、环境和社会科学等相关数据经常出现的领域产生重大影响和许多应用。特别是，QIF方法大大改进了广义估计方程设置中回归参数的估计（Liang & Zeger, 1986）。考虑到一个用于健康影响评估的空气污染的现实例子，回归参数估计值即使有微小的差异，也会对我们的健康和环境政策产生重大影响。此外，它也是第一次将计量经济学中的广义矩量方法（Hansen, 1982）与统计领域的估计函数联系起来。它试图回答计量经济学家经常提出的一个问题：如何以尽可能低的维度选择信息量最大的矩条件。该研究还将通过开发一门关于纵向数据和研究生培训的新课程来服务于教育目的。

This research focuses on a new statistical method for the analysis of correlated data, the quadratic inference function (QIF) approach (Qu, Lindsay & Li 2000). The QIF is built on a semiparametric framework defined by a set of mean zero estimating functions, but differs from the standard estimating function approach in that there are more equations than the number of unknown parameters. The QIF has advantages compared to the estimating function approach, such as not requiring the specification of the likelihood function. It also overcomes limitations of the estimating function approach such as a lack of objective functions and likelihood functions for testing. One of the main goals of the proposed project is to explore the QIF for robustness with respect to the consistency of estimators when mean zero assumptions are not satisfied. A second goal focuses on the missing data problem, which occurs often in longitudinal data. Testing whether missing data are ignorable is still a challenging problem in general. The goodness-of-fit test for the QIF appears to be a valid test for nonignorable missing data. The third goal is to test order restricted alternative hypotheses for correlated data using the QIF. Current existing testing tools are not satisfactory and are mainly based on the likelihood function for parametric models, and therefore are not applicable for correlated data where the likelihood function is difficult to formulate. The QIF is related to the empirical likelihood (Owen, 1988) which is popular for nonparametric models. The proposed project also illustrates the Edgeworth expansion of QIF and explores how to apply the bootstrap strategy to improve testing accuracy for small samples of correlated data.This research will have a significant impact and many applications in biostatistics, econometrics, and the environmental and social sciences where correlated data arise often. In particular, the QIF method substantially improves the estimation of regression parameters in generalized estimating equation settings (Liang & Zeger, 1986). Considering a real world example of air pollution for health impact assessment, even a slight difference in the regression parameter estimates can have a major impact on our health and environmental policies. Further, it is also the first effort to connect the generalized method of moments (Hansen, 1982) in econometrics to estimating functions in the statistics field. It attempts to answer a question frequently asked by econometricians: how to choose the most informative moment conditions with the lowest dimension possible. The research will also serve an educational purpose through developing a new course on longitudinal data and training of graduate students.