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Theorem 3 tells us that δg0 (X), with g0 (|X|) = min(m tanh(m|X|), g(|X|)), dominates δg (X). Here are some additional observations related to the previous example (but also applicable to the general case of this section). (i) The set A|x| = [0, m tanh(m|x|)] can be interpreted as yielding a complete class of invariant estimators with the upper envelope corresponding to the Bayes estimator δBU (X) associated with the uniform prior on {−m, m}. (ii) In Example 2, the dominating estimator δg0 (X) of Theorem 3 will be given by the Bayes estimator δBU (X) if and only if m tanh(m|x|) ≤ g(|x|), for all x.

A method for evaluating improper prior distributions. In Statistics Decision Theory and Related Topics III (S. S. Gupta and J. O. ) 1, 329–352. New York:Academic Press. 20 M. L. Eaton [9] Eaton, M. L. (1992). A statistical diptych: Admissible inferences- Recurrence of symmetric Markov chains. Ann. Statist. 20, 1147–1179. MR1186245 [10] Eaton, M. L. (2001). Markov chain conditions for admissibility with quadratic loss. In State of the Art in Statistics and Probability, a Festschrift for Willem van Zwet edited by M.

Note that the above defined Y1 and Y2 are independently normally distributed σ12 1 −θ2 1 +θ2 (with E[Y1 ] = µ1 = θ1+τ , E[Y2 ] = µ2 = τ θ1+τ , Cov (Y1 ) = 1+τ Ip , and Cov (Y2 ) = τ σ12 1+τ Ip ). Given this independence, the risk function of δφ (for θ = (θ1 , θ2 )) becomes R θ, δφ (X1 , X2 ) = Eθ = Eθ Y2 − = Eθ Y2 − µ2 Y2 + φ(Y1 ) − θ1 τ θ 1 + θ2 1+τ 2 2 θ1 − θ2 1+τ + φ(Y1 ) − + Eθ φ(Y1 ) − µ1 2 2 . Therefore, the performance of δφ (X1 , X2 ) as an estimator of θ1 is measured solely by the performance of φ(Y1 ) as an estimator of µ1 under the model Y1 ∼ σ12 Np (µ1 , 1+τ Ip ), with the restriction µ1 ∈ C = {y : (1 + τ )y ∈ A}.

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