A Festschrift for Herman Rubin (Institute of Mathematical - download pdf or read online

By Anirban Dasgupta

ISBN-10: 0940600617

ISBN-13: 9780940600614

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

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Much of what follows has Katz’s result as an examplar. A great deal of the material in sections 2, 3 and 5 is focussed on extending aspects of Katz’s result. If, in the above normal example, the restricted space is a compact interval, then the projection of the usual estimator still dominates the unrestricted MLE but cannot be minimax for quadratic loss because it is not Bayes. In this case Casella and Strawderman (1981) and Zinzius (1981) showed that the unique minimax estimator of the mean θ for a restriction of the form θ ∈ [−m, m] is the Bayes estimator corresponding to a 2 point prior on {−m, m} for m sufficiently small.

And Roynette, B. (1977). March aleatoires sur les Groupes de Lie. Lecture Notes in Math 624, New York: Springer. MR517359 [13] Hobert, J. and Robert, C. (1999). Eaton’s Markov chain, its conjugate partner, and P -admissibility. Ann. 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.

1. Example Here is an illustration of Kubokawa’s IERD method for an observable X generated from a location family density fθ (x) = f0 (x − θ), with known f0 , where Eθ [X] = θ, and Eθ [X 2 ] < ∞. For estimating θ, with squared error loss (d − θ)2 , under the constraint θ ≥ a (hereafter, we will take a = 0 without loss of generality), we show that the Generalized Bayes estimator δU (X) with respect to the uniform prior π(θ) = 1(0,∞) (θ) dominates the MRE estimator δ0 (X) = X. As a preliminary to the following dominance result, observe that δU (X) = X + hU (X), where hU (y) = ∞ (θ − y)f0 (y − θ) dθ 0 ∞ f0 (y − θ) dθ 0 − = y uf0 (u) du −∞ y f (u) du −∞ 0 = −E0 [X|X ≤ y]; and that hU is clearly continuous, nonincreasing, with hU (y) ≥ − limy→∞ E0 [X|X ≤ y] = −E0 [X] = 0.

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

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