Regression and Deconvolution with Heteroscedastic Measurement Error

在这项研究的第一部分，“具有重复测量的异方差测量误差的无偏估计和校正计分方法”，研究人员开发了一种当数据测量有误差时进行统计推断的一般方法。从假设对无误差的数据已知有效的统计估计方法，以及对容易出错的变量进行重复测量的假设出发，研究人员展示了如何修改通常的估计方法以消除测量误差引起的偏差。关键的技术进步包括容纳异方差测量误差和重复测量，以及发展一种新的蒙特卡罗方法，对正常均值进行无偏估计。作者将一般方法应用于m-估计和密度估计，从而包含了广泛的统计推断问题。在这项研究的第二部分，“用辅助数据去卷积”，研究人员探索了去卷积问题的方法，即利用与用误差测量的变量相关的辅助变量。辅助变量的作用类似于工具变量，用于减少反卷积估计中的变异性。天文学家对星系距离的测量，流行病学家对受试者血压的测量，环境科学家对每日空气污染水平的测量，以及社会学家对受试者行为和态度的测量，都有一个共同的事实，即所有这些都不是完全准确的。测量误差是跨越学科界限的数据分析和解释中普遍存在的问题。它是不确定性的来源，可能会对从数据得出的估计产生偏差，并导致错误的推断。在这个项目中，研究人员开发了当测量数据有误差时的统计推断的理论和方法。这项研究为统计推断中的一个长期存在的问题提供了一种新的解决方案，并使用该解决方案为分析有误差的测量数据提供了一种全面的方法。主要的好处是改进了统计推断，其形式是根据科学数据计算出的更少偏见和更准确的估计。由于有误差测量的数据普遍存在，这项研究的影响也将同样广泛，不仅会立即应用于上述科学领域，而且还会应用于其他许多领域。

In the first component of the research, "Unbiased Estimation and Corrected-Score Methods for Heteroscedastic Measurement Error with Replicate Measurements," the investigator develops a general approach to statistical inference when data are measured with error. Starting with the assumptions that a valid statistical estimation method is known for error-free data, and that replicate measurements are made of the error-prone variate, the investigator shows how to modify the usual estimation method to eliminate bias induced by measurement error. The key technical advances include the accommodation of heteroscedastic measurement errors and replicate measurements, as well as development of a new Monte Carlo method of unbiased estimation of a normal mean. The author applies the general approach to m-estimation and density estimation, thereby incorporating a broad scope of statistical inference problems. In the second component of the research, "Deconvolution with Auxiliary Data," the investigator explores approaches to the deconvolution problem that exploit auxiliary variables correlated to the variable measured with error. The auxiliary variables play a roll akin to that of instrumental variables and are used to reduce variability in the deconvolution estimates.The astronomer's measurements of distances to galaxies, the epidemiologist's measurements of subjects' blood pressures, the environmental scientist's measurements of daily air pollution levels, and the sociologist's measurements of subjects' behaviors and attitudes share in common the fact that all are less than perfectly accurate. Measurement error is a pervasive problem in the analysis and interpretation of data that crosses disciplinary boundaries. It is a source of uncertainty that can bias estimates derived from data and lead to erroneous inferences. In this project the investigator develops theory and methods for statistical inference when data are measured with error. The research provides a new solution to a long-standing problem in statistical inference, and uses that solution to provide a comprehensive approach to the analysis of data measured with error. The primary benefit is improved statistical inference in the form of less biased and more accurate estimates calculated from scientific data. Because the prevalence of data measured with error is widespread, the impact of the research will be similarly widespread, finding immediate applications not only to the scientific fields mentioned above, but numerous others as well.