U.S.-Australia Cooperative Research: Topics in Semiparametric Modeling

该奖项将支持康奈尔大学的大卫·鲁珀特博士和澳大利亚国立大学的彼得·霍尔博士之间的合作研究。二元回归分析的问题是将结果Y与预测因子X联系起来。这种问题的一个例子是将饮食和其他个人因素(X)联系起来以预测乳腺癌的发展(Y)。通常，这种二元回归问题是通过Logistic回归关系来建模的。本文研究的重点是当X有测量误差时，用一个替代物W作为X的估计值的情况。最近，数学家们研究了预测值中带有测量误差的Logistic回归问题。并指出，忽略测量误差，对Y对W进行普通的Logistic回归，会导致对这些方程中参数的不一致和严重偏差的估计。一种解决测量误差问题的方法是半参数方法，它依赖于原始的参数Logistic回归模型和非参数回归技术。在这一拟议解决方案的基础上，该项目的目标是通过将经典参数技术与更现代的非参数技术相结合，解决对统计学具有普遍重要性的几个问题。涉及非参数分量和参数分量估计的理论问题将被研究。结合参数和非参数技术来解决预报器受到测量误差影响时的二元回归问题，将进行研究。第三个研究领域将涉及数据转换和非参数回归的结合。该项目代表了这位美国数学家和澳大利亚团队之间的出色合作，前者在国家科学基金会的支持下，专注于非参数回归分析，后者发表了这一领域的许多开创性论文。这是推进非参数技术在数理统计重要问题中的应用的重要基础性工作。

This award will support collaborative research between Dr. David Ruppert of Cornell University and Dr. Peter Hall of the Australian National University. The problem of binary regression analysis is to relate an outcome Y to a predictor X. An example of such a problem is to relate diet and other personal factors (X) to predict the development of breast cancer (Y). Usually such binary regression problems are modeled through a logistic regression relationship. The focus of this research is on cases when X has measurement error and a surrogate W is used as an estimate of X. Recently, mathe- maticians have studied the problem of logistic regression with measurement error in predictors, and have pointed out that ignoring the measurement error and performing an ordinary logistic regression of Y on W leads to inconsistent and seriously biased estimates of the parameters in these equations. One solution which has been proposed to the problem of measurement error is a semiparametric method, which relies on the original parametric logistic regression model combined with nonparametric regression techniques. Building on this proposed solution, the objective of this project is to work on several problems of general importance to statistics through combining classical parametric with the more modern nonparametric techniques. Theoretical problems involving estimation of nonparametric and parametric components will be investigated. Combination of parametric and nonpara- metric techniques to solve the problem of binary regression when the predictors are subject to measurement error will be studied. A third study area will concern combining data transformation and nonparametric regression. The project represents excellent collaboration between the U.S. mathematician, whose work, under National Science Foundation support, focuses on nonparametric regression analysis and the Australian group, who have published many of the seminal papers in this area. This is important fundamental work in advancing the use of nonparametric techniques in important problems of mathematical statistics.