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On minimax statistical procedures and their admissibility. Ann. Math. Statist. 22, 22–42. MR39966 [3] Brown, L. (1971). Admissible estimators, recurrent diffusions, and insolvable boundary value problems. Ann. Math. Statist. 42, 855–904. MR286209 [4] Brown, L. (1979). A heuristic method for determining admissibility of estimators – with applications. Ann. Statist. 7, 960–994. MR536501 [5] Brown, L. D. and Hwang, J. T. (1982). A unified admissibility proof. In Statistical Decision Theory and Related Topics III (S.

Let C be any λ-proper set so λ(C) < +∞ and let Ei = Ci ∩ C, i = 1, 2, . . 1. Since Ei C and λ(C) < +∞, we have lim λ(Ei ) −→ λ(C) i−→∞ and lim λ(C ∩ Eic ) −→ 0. 3) yields 1 1 1 H 2 (C) ≤ H 2 (Ei ) + 2 2 λ(C ∩ Eic ). The right hand side of this inequality converges to zero as i −→ ∞. Hence H(C) = 0. Since C was an arbitrary λ-proper set, the chain W is locally-ν-recurrent. Acknowledgment Many thanks to Jim Hobert, Tiefeng Jiang and Galin Jones for their valuable comments. Also special thanks to Anirban Das Gupta for his efforts on this Festschrift for Herman Rubin and his many comments on this contribution.

This is so because, as remarked upon in Section 2, φ1 (Y1 ) = φmle (Y1 ) dominates under squared error loss, as an estimator of µ1 ; µ1 ∈ C; φ0 (Y1 ) = Y1 . Observe further that the maximum likelihood estimator δmle (X1 , X2 ) of θ1 for the parameter space Θ = {(θ1 , θ2 ) : θ1 − θ2 ∈ A, τ θ1 + θ2 ∈ p } is indeed given by: δmle (X1 , X2 ) = (µˆ2 )mle + (µˆ1 )mle = Y2 + φmle (Y1 ), given the independence and normality of Y1 and Y2 , and the fact that Y2 is the MLE of µ2 (µ2 ∈ p ). Our next two applications of Proposition 1 deal with the estimator δmle (X1 , X2 ).

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