Improved Multiple Confidence Intervals via Thresholding Informed by Prior Information
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Consider a statistical problem where a set of parameters are of interest to a researcher. Then multiple confidence intervals can be constructed to infer the set of parameters simultaneously. The constructed multiple confidence intervals are the realization of a multiple interval estimator (MIE), the main focus of this study. In particular, a thresholding approach is introduced to improve the performance of the MIE. The developed thresholds require additional information, so a prior distribution is assumed for this purpose. The MIE procedure is then evaluated by two performance measures: a global coverage probability and a global expected content, which are averages with respect to the prior distribution. The procedure defined by the performance measures will be called a Bayes MIE with thresholding (BMIE Thres). In this study, a normal-normal model is utilized to build up the BMIE Thres for a set of location parameters. Then, the behaviors of BMIE Thres are investigated in terms of the performance measures, which approach those of the corresponding z-based MIE as the thresholding parameter, C, goes to infinity. In addition, an optimization procedure is introduced to achieve the best thresholding parameter C. For illustrations, in-season baseball batting average data and leukemia gene expression data are used to demonstrate the procedure for the known and unknown standard deviations situations, respectively. In the ensuing simulations, the target parameters are generated from different true generating distributions to consider the misspecified prior situation. The simulation also involves Bayes credible MIEs, and the effectiveness among the different MIEs are compared with respect to the performance measures. In general, the thresholding procedure helps to achieve a meaningful reduction in the global expected content while maintaining a nominal level of the global coverage probability.
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