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Is dominating Positive Part James Stein estimator when estimating the mean of a multivariate normal of dimension 3 with known variance(all equal) an open problem?

If not, what is this estimator called? Can you point out some references?

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Yes, there is a published reply to this challenge, by Peter Yi-Shi Shao and William E. Strawderman in 1994: https://projecteuclid.org/euclid.aos/1176325640

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They provide conditions on estimators of the type

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to ensure domination. Interestingly such estimators are also inadmissible and hence are dominated by other (generalised Bayes) estimators (see p. 1530).

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  • $\begingroup$ Do we know the admissible Generalized Bayes estimators that dominate JS, or do we just rely on some existence theorem? $\endgroup$ – Cagdas Ozgenc Oct 31 '18 at 10:37
  • $\begingroup$ No, as mentioned on p.1530, there exist generalised Bayes estimators dominating this one, they may belong to the class discussed on p. 1530, if not to (3.1) [which does not contain analytic functions], but no explicit improvement is known. $\endgroup$ – Xi'an Oct 31 '18 at 16:55
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    $\begingroup$ By the sufficiency principle, one should only consider the sample mean. $\endgroup$ – Xi'an Nov 1 '18 at 11:51

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