Mathematical Sciences: Measurement Error and Statistical Inference

本研究发展了对有误差测量数据进行统计推断的新方法。这些方法利用高速计算的可用性来模拟测量误差对数据集的影响。这种模拟实验的结果构成了一种称为模拟外推的推理方法的基础。研究的第一部分将模拟外推估计方法应用于对误差测量的变量进行重复测量的情况。研究的第二部分使用模拟外推的复杂变量公式来绕过外推步骤。这是通过使用复杂伪随机变量执行模拟实验来计算感兴趣参数的无偏估计方程来完成的。研究的第三个组成部分侧重于双样本假设检验问题，开发当一个样本的数据测量有误差时渐近有效的检验。数据中测量误差的存在会影响对这些数据进行统计分析所得结论的有效性。例如，血压或胆固醇水平等危险因素的测量误差会影响心脏病与这些危险因素之间关系的统计确定，通常会导致对控制这些危险因素的有益影响的低估。同样，对生态系统健康与生态系统压力源（例如污染物）之间关系的统计确定也可能因无法准确测量压力源水平而模糊不清。本研究发展了统计理论和方法，以便从有误差的测量数据中做出有效的推断。测量误差发生在许多科学领域（例如调查方法、环境研究、流行病学、医学），这一事实意味着这项研究具有广泛的适用性。该研究使用新颖的计算机模拟方法来模拟测量误差对数据集的影响以及对从这些数据中得出的结论的影响。从模拟研究中获得的信息然后用于从没有测量误差的偏置影响的数据中做出有效的推断。

This research develops new methods of statistical inference for data measured with error. The methods exploit the availability of high-speed computing to simulate the effects of measurement error on a data set. The results of such simulation experiments form the basis of a method of inference called simulation extrapolation. The first component of the research adapts the method of simulation extrapolation estimation to situations in which replicate measurements are made on the variables measured with error. The second component of the research uses the complex-variable formulation of simulation extrapolation to bypass the extrapolation step. This is accomplished by performing a simulation experiment using complex pseudo random variables to compute unbiased estimating equations for the parameters of interest. The third component of the research focuses on two-sample hypothesis testing problems, developing tests that are asymptotically valid when data from one sample are measured with error. The presence of measurement error in data can affect the validity of conclusions drawn from statistical analyses of such data. For example, measurement error in risk factors such as blood pressure or cholesterol level affects statistical determination of the relationship of heart disease to such risk factors, usually resulting in underestimation of the beneficial effects of controlling these risk factors. Similarly, statistical determination of the relationship of ecosystem health to ecosystem stressers (pollutants, for example) can be obscured by the inability to accurately measure stresser levels. This research develops statistical theory and methods for making valid inferences from data that are measured with error. The fact that errors of measurement occur in many areas of science (survey methodology, environmental studies, epidemiology, medicine for example) means that the research is widely applicable. The research uses novel computer simulation methods to simulate the effects of measurement error on a data set and on the conclusions drawn from such data. The information obtained from the simulation studies is then used to make valid inferences from the data that are free from the biasing effects of measurement error.