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.
