A sharp necessary and sufficient condition for inadmissibility of estimators in a control problem
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 [\textit{J. O. Berger, L. M. Berliner} and \textit{A. Zaman,} ibid. 10, 838-856 (1982; Zbl 0496.62005)] are given in section 5.
orthogonally invariant nonrandomized decision rules
scientific article; zbMATH DE number 3872461