The Empirical Bayes Approach For Shrinkage Confidence Intervals

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The parametric empirical Bayes, introduced by [12], [13], [14], and [34], is gaining more and more attention in both the theoretic and applied statistics. Its ability in borrowing strength makes this idea prevalent in modern technology such as microarray where the number of parameters is very large and the number of observations for each parameter is much smaller. In this dissertation, we will apply this idea into constructing confidence interval for different models and problem settings. In Chapter 2, we introduce the Log-Normal model and construct the empirical Bayesian confidence interval for each parameter by shrinking both means and variances for the very first time. Keeping the coverage probability above the nominal level, the new construction enjoys the shortest average length among all the confidence interval constructed as demonstrated by extensive numerical study as well as in a real data set where the real parameters are known. In Chapter 3, we deal with the simultaneous interval construction, where the criterion of controlling the simultaneous coverage probability appears to be too conservative. We propose and study the control of the empirical Bayes False Coverage Rate (FCR) to address the multiplicity. We further construct intervals which controls the empirical Bayesian FCR under the normal-normal model. In Chapter 4, we deal with the model with mixed prior which is more practical in microarray technology and construct intervals which can control the empirical Bayesian FCR. All the procedures we have derived in this work based on the empirical Bayes approach are explicitly defined and can be computed instantaneously.
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