Estimating Equations and Second-Order Theories

DMS 96-266249 Li本研究集中在与二阶理论有关的两个问题上：一个是精度的评估，另一个是估计方程的精化。估计的准确性部分取决于估计者的质量，部分取决于偶然性。传统上，机会因素是通过条件或贝叶斯方法来考虑的，这需要辅助或先验分布。然而，机会元素可以在没有这两者的情况下存在。研究的第一部分通过渐近展开和几何分析直接估计损失来解决这个问题。在这个方向上获得的一个特殊结果是，倒置的观测信息最接近平方误差。研究的第二部分是关于提高估计方程的二阶精度。之前提出了两种方法，但它们不适用于激励本项目的一种重要类型的情况。此外，根据估计方程与经典极大似然估计的二阶性质的相似性，研究了估计方程的几个以前未知的性质。这项研究将对半参数方法有更深入的理解和更有效的使用。科学的最新进展，特别是在医学、生物学、社会学和生态学研究方面的进展，极大地增加了数据集的规模和复杂性。随着计算机能力的不断增强，这一变化向传统的统计方法提出了新的挑战。这些挑战带来的一个研究领域是估计方程，这是本研究的重点。估算方程允许科学家直接对最感兴趣的参数进行建模，而不会像传统方法那样做过多的假设，因为这种假设会影响对参数的推断。估计方程对于具有复杂依赖结构的数据集特别有用，例如纵向研究和散布在自然环境中的植物的研究。在过去的几十年里，估计方程的研究取得了长足的进步，然而，大多数研究涉及的是可以被称为粗略水平的方面(或一阶方面)。然而，对更精细的方面(或二阶性质)的研究才刚刚开始。本文系统地研究了估计方程的二阶问题。

DMS 96-266249 Li The research is focused on two problems related to second order theories: one on the assessment of accuracy and the other on the refinement of estimating equations. The accuracy of an estimate is determined partly by the quality of the estimator and partly by chance. Traditionally the chance element was accounted for by the conditional or the Bayesian approach, which require an ancillary or a prior distribution. However, the chance element can exist without either. The first part of the research tackles this problem by direct estimation of loss via asymptotic expansions and geometric analyses. A particular result obtained in this direction is that the inverted observed information best approximates the squared error. The second part of the research is concerned with improving the second-order accuracy of an estimating equation. Two methods were previously proposed, but they are inapplicable to an important type of situations, which motivates this project. In addition, several previously unknown properties of estimating equations are investigated in the light of their analogy with the second-order properties of the classical maximum likelihood estimator. This research will yield deeper understanding and more effective use of semiparametric methods. %%% The recent advances in science, particularly in medical, biological, sociological, and ecological studies, have drastically increased the scale and complexity of data sets. This change, hand in hand with the ever increasing computer power, gives new challenges to traditional statistical methodologies. One area of studies that these challenges have brought about is that of estimating equations, which is the focus of the present research. Estimating equations allow scientists to model directly the parameters which are of the most interest without making excessive assumptions (as traditional methods often do), whose violat ions would impair the inference about the parameters. Estimating equations are especially useful for data sets with complicated dependence structures, such as longitudinal studies and the studies of plants scattered in a natural environment. The studies of estimating equations have undergone vigorous advances during the past decades, most of which, however, are concerned with what might be called the coarser-level aspects (or first-order aspects). Whereas the studies of the finer-level aspects (or second-order properties) have just begun to catch up. The second-order aspects of estimating equations are systematically investigated in the present research.