Statistical Analysis in Empirical Bayes and in Causal inference

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Doctor of Philosophy (PhD)
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Statistical Theory
Statistics and Probability
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In Part I titled Empirical Bayes Estimation, we discuss the estimation of a heteroscedastic multivariate normal mean in terms of the ensemble risk. We first derive the ensemble minimax properties of various estimators that shrink towards zero through the empirical Bayes method. We then generalize our results to the case where the variances are given as a common unknown but estimable chi-squared random variable scaled by different known factors. We further provide a class of ensemble minimax estimators that shrink towards the common mean. We also make comparison and show differences between results from the heteroscedastic case and those from the homoscedastic model. In Part II titled Causal Inference Analysis, we study the estimation of the causal effect of treatment on survival probability up to a given time point among those subjects who would comply with the assignment to both treatment and control when both administrative censoring and noncompliance occur. In many clinical studies with a survival outcome, administrative censoring occurs when follow-up ends at a pre-specified date and many subjects are still alive. An additional complication in some trials is that there is noncompliance with the assigned treatment. We first discuss the standard instrumental variable method for survival outcomes and parametric maximum likelihood methods, and then develop an efficient plug-in nonparametric empirical maximum likelihood estimation (PNEMLE) approach. The PNEMLE method does not make any assumptions on outcome distributions, and makes use of the mixture structure in the data to gain efficiency over the standard instrumental variable method. Theoretical results of the PNEMLE are derived and the method is illustrated by an analysis of data from a breast cancer screening trial. From our limited mortality analysis with administrative censoring times 10 years into the follow-up, we find a significant benefit of screening is present after 4 years (at the 5% level) and this persists at 10 years follow-up.

Lawrence Brown
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