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New PDF release: A Festschrift for Herman Rubin (Institute of Mathematical

By Anirban Dasgupta

ISBN-10: 0940600617

ISBN-13: 9780940600614

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Additional info for A Festschrift for Herman Rubin (Institute of Mathematical Statistics, Lecture Notes-Monograph Series)

Sample text

M − g(x)| is “small” for some x). , the maximal invariant) |X| to obtain (below, the notation Eθ represents the expectation with respect to |X|) X −θ |X| 2 |X| Eθ |X| θ2 + g 2 (|X|) − 2Eθ = Eθ |X| θ2 + g 2 (|X|) − 2g(|X|)A|X|(θ) , = θEθ X |X| = θ |X| R(θ, δg (X)) = Eθ = Eθ g(|X|) |X| θX g(|X|) |X| |X| where A|X| (θ) f (|X| − θ) − f (|X| + θ) f (|X| − θ) + f (|X| + θ) , (as in (6) below) by symmetry of f . Now, rewrite the risk as |X| R(θ, δg (X)) = Eθ |X| θ2 − A2|X| (θ) + Eθ g(|X|) − A|X| (θ) 2 , (3) to isolate with the second term the role of g, and to reflect the fact that the performance of the estimator δg (X), for θ ∈ [−m, m], is measured by the average distance 2 g(|X|) − A|X| (θ) under f (x − θ).

Statist. 27, 361–373. MR1701115 [14] James, W. and Stein, C. (1961). Estimation with quadratic loss. In Proc. Fourth Berkeley Symp. Math. Statist. Probab. 1, 361–380. Berkeley: University of California Press. MR133191 [15] Johnson, B. (1991). On the admissibility of improper Bayes inferences in fair Bayes decision problems. D. Thesis, University of Minnesota. [16] Johnstone, I. (1984). Admissibility, difference equations and recurrence in estimating a Poisson mean. Ann. Statist. 12, 1173–1198. MR760682 [17] Johnstone, I.

Lemma 1. If δπ is a Bayes estimator with respect to a prior distribution π, and Sπ = {θ ∈ Θ : supθ {R(θ, δπ )} = R(θ, δπ )}, then δπ is minimax whenever Pπ (θ ∈ Sπ ) = 1. Casella and Strawderman’s work capitalized on Karlin’s (1957) sign change arguments for implementing Lemma 1 while, in contrast, the sufficient conditions E. Marchand and W. E. Strawderman 34 obtained by Zinzius concerning the minimaxity of δBU (X) were established using the “convexity technique” as stated as part (b) of the following Corollary to Lemma 1.

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A Festschrift for Herman Rubin (Institute of Mathematical Statistics, Lecture Notes-Monograph Series) by Anirban Dasgupta


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