harvard university · 2017. 2. 12. · the optimal confidence region for a random parameter hajime...

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Harvard University Harvard University Biostatistics Working Paper Series Year Paper The Optimal Confidence Region for a Random Parameter Hajime Uno * Lu Tian L.J. Wei * Harvard University, [email protected] Northwestern University, [email protected] Harvard University, [email protected] This working paper is hosted by The Berkeley Electronic Press (bepress) and may not be commer- cially reproduced without the permission of the copyright holder. http://biostats.bepress.com/harvardbiostat/paper13 Copyright c 2004 by the authors.

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Page 1: Harvard University · 2017. 2. 12. · The Optimal Confidence Region for a Random Parameter Hajime Uno, Lu Tian, and L.J. Wei Abstract Under a two-level hierarchical model, suppose

Harvard UniversityHarvard University Biostatistics Working Paper Series

Year Paper

The Optimal Confidence Region for a RandomParameter

Hajime Uno∗ Lu Tian†

L.J. Wei‡

∗Harvard University, [email protected]†Northwestern University, [email protected]‡Harvard University, [email protected]

This working paper is hosted by The Berkeley Electronic Press (bepress) and may not be commer-cially reproduced without the permission of the copyright holder.

http://biostats.bepress.com/harvardbiostat/paper13

Copyright c©2004 by the authors.

Page 2: Harvard University · 2017. 2. 12. · The Optimal Confidence Region for a Random Parameter Hajime Uno, Lu Tian, and L.J. Wei Abstract Under a two-level hierarchical model, suppose

The Optimal Confidence Region for a RandomParameter

Hajime Uno, Lu Tian, and L.J. Wei

Abstract

Under a two-level hierarchical model, suppose that the distribution of the ran-dom parameter is known or can be estimated well. Data are generated via a fixed,but unobservable realization of this parameter. In this paper, we derive the small-est confidence region of the random parameter under a joint Bayesian/frequentistparadigm. On average this optimal region can be much smaller than the cor-responding Bayesian highest posterior density region. The new estimation pro-cedure is appealing when one deals with data generated under a highly parallelstructure, for example, data from a trial with a large number of clinical centersinvolved or genome-wide gene-expession data for estimating individual gene- orcenter-specific parameters simultaneously. The new proposal is illustrated with atypical microarray data set and its performance is examined via a small simulationstudy.

Page 3: Harvard University · 2017. 2. 12. · The Optimal Confidence Region for a Random Parameter Hajime Uno, Lu Tian, and L.J. Wei Abstract Under a two-level hierarchical model, suppose

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Page 4: Harvard University · 2017. 2. 12. · The Optimal Confidence Region for a Random Parameter Hajime Uno, Lu Tian, and L.J. Wei Abstract Under a two-level hierarchical model, suppose

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Page 5: Harvard University · 2017. 2. 12. · The Optimal Confidence Region for a Random Parameter Hajime Uno, Lu Tian, and L.J. Wei Abstract Under a two-level hierarchical model, suppose

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-1 -0.5 0 0.5 1 1.5 2 2.5 3-2

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http://biostats.bepress.com/harvardbiostat/paper13