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

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An interesting open problem is to find admissible minimax estimators in this setting which do not have the same risk at θ = 0 as the unrestricted MLE, and, in particular, to find an admissible minimax estimator dominating the restricted MLE. We will concern ourselves primarily with methods that have proven to be successful in such problems, and somewhat less so with cataloguing the vast collection of results that have appeared in the literature. In particular, we will concentrate on the following methods.

6) yields 1 1 1 1 H 2 (B) ≤ ∆ 2 (h) + 2 2 λ 2 B ∩ Ac . 3) holds. This completes the proof. 4) follows. For h1 and h2 in L2 (λ), consider the symmetric bilinear form < h1 , h2 >= h1 (y)h2 (y)λ(dy) − h1 (y)h2 (z)R(dy|z)λ(dz). That < ·, · > is non-negative definite is a consequence of the symmetry of R and the Cauchy–Schwarz inequality: 2 h1 (y)h1 (z)R(dy|z)λ(dz) ≤ h21 (y)R(dy|z)λ(dz) · 2 = h21 (y)λ(dy) . h21 (z)R(dy|z)λ(dz) Evaluating improper priors and the recurrence of symmetric Markov chains 19 1 Thus h =< h, h > 2 is a semi-norm on L2 (λ) and so is subadditive.

On the distribution of values of sums of random variables. Mem. Amer. Math. Soc. 6, 1–12. MR40610 [7] Diaconis, P. and Strook, D. (1991). Geometric bounds for eigenvalues of Markov chains. Am. Appl. Probab. 1, 36–61. MR1097463 [8] Eaton, M. L. (1982). 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.

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