Regression and Deconvolution with Heteroscedastic Measurement Error

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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.