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By Dasgupta A. (ed.)

ISBN-10: 0940600617

ISBN-13: 9780940600614

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1) h∈V (C) Also, let V ∗ (C) = {h|h ∈ V (C), h(y) ∈ [0, 1] for y ∈ C c }. The results in Appendix 2 of Eaton (1992) show that H(C) = inf ∆(h). 1. Consider measurable subsets A and B of Y that are both λ-proper. If A ⊆ B, then 1 1 1 1 1 H 2 (A) ≤ H 2 (B) ≤ H 2 (A) + 2 2 λ 2 (B ∩ Ac ). 3) M. L. Eaton 18 Proof. 3) is 1 obvious. For the right hand inequality, first note that ∆ 2 is a subadditive function defined on L2 (λ)– that is, 1 1 1 ∆ 2 (h1 + h2 ) ≤ ∆ 2 (h1 ) + ∆ 2 (h2 ). 4) is given below. 5) for h1 and h3 in L2 (λ).

We wish to estimate θ1 with squared error loss δ − θ1 2 when we know for example that θ1 −θ2 ∈ A for some set A. We illustrate the application of a rotation technique, used perhaps first by Blumenthal and Cohen (1968a), as well as Cohen and Sackrowitz (1970), which, loosely described, permits to subdivide the estimation problem into parts that can be separately handled. Section 6 deals with minimaxity, and particularly those results related to Casella and Strawderman (1981) and Zinzius (1981) establishing minimaxity of Bayes estimators relative to 2 point priors on the boundary of a sufficiently small one dimensional parameter space of the form [a, b].

Therefore the above theory applies to the Markov chain W = (W0 , W1 , W2 , . ) on Θ∞ defined by R(dθ|η). 1” stated in the introductory section of this paper. 4. Suppose the Markov chain W with state space Θ and transition function R(dθ|η) is locally-ν-recurrent. 2) is strongly-admissible. Proof. 7) is zero for each νproper set C. 3 holds. Thus, Q(dθ|x) is strongly admissible. 3 provides an extremely useful sufficient condition for this property of W . An application is given in the next section.

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A festschrift for Herman Rubin by Dasgupta A. (ed.)

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