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A sharp necessary and sufficient condition for inadmissibility of estimators in a control problem. (English) Zbl 0547.62006
Let a standard normal model of the control problem $$Y=\theta^ tz+\epsilon$$ be given, where $$\theta$$ has to be estimated with respect to the loss $$L(\theta,\delta)=(1-\theta^ t\delta)^ 2$$. An estimator for $$\delta$$ can be evaluated by its risk function $$R(\theta$$,$$\delta)$$, and is called inadmissible if there exists $$\delta^*$$ such that $$R(\theta,\delta^*)\leq R(\theta,\delta)$$ for all $$\theta$$ with strict inequality for some $$\theta$$.
This paper deals with orthogonally invariant nonrandomized decision rules of the form $$\delta (x)=\phi (| x|)x/| x|$$. Generalized Bayes estimators with respect to spherically symmetric priors have such a representation.
After three preparatory lemmas in section 2, a necessary condition (theorem 3.1) and a sufficient condition (theorem 4.1) for the mentioned generalized Bayes estimators are shown. Applications (theorems 5.1, 5.2) and comparisons with different approaches of the admissibility concept [J. O. Berger, L. M. Berliner and A. Zaman, ibid. 10, 838-856 (1982; Zbl 0496.62005)] are given in section 5.
Reviewer: F.Fahrion

##### MSC:
 62C15 Admissibility in statistical decision theory 62F15 Bayesian inference 62F10 Point estimation 62P20 Applications of statistics to economics
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