# Download A Festschrift for Herman Rubin (Institute of Mathematical by Anirban Dasgupta PDF

By Anirban Dasgupta

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Marchand and W. E. 3. Some related work Interestingly, the dominance result in (iii) of the normal model MLE was previously established, in a diferent manner, by Casella and Strawderman (1981) (see also Section 6). As well, other dominating estimators here were provided numerically by Kempthorne (1988). For the multivariate version of Example 2: X ∼ Np (θ, Ip ); (p ≥ 1); with θ ≤ m, Marchand and Perron (2001) give dominating estimators of δmle (X) under squared error loss d−θ 2 . Namely, using a similar risk decomposition as above, including argument (ii), they show that δBU (X) (Bayes with respect to a boundary √ uniform prior) dominates δmle (X) whenever m ≤ p.

For applications of Lai’s ideas to the multivariate Poisson case, we refer the reader to Lai’s thesis. 1. 2. To this end, consider a measurable subset C ⊆ Y that is λ-proper and let H(C) = inf ∆(h). 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.

If δπ is a Bayes estimator with respect to a two-point prior on {a, b} such that R(a, δπ ) = R(b, δπ ), then δπ is minimax for the parameter space Θ = [a, b] whenever, as a function of θ ∈ [a, b], (a) ∂ ∂θ R(θ, δπ ) has at most one sign change from − to +; or (b) R(θ, δπ ) is convex. Although the convexity technique applied to the bounded normal mean problem gives only a lower bound for m0 ; (Bader and Bischoff (2003) report that the best √ known bound using convexity is 22 , as given by Bischoff and Fieger (1992)); it has proven very useful for investigating least favourable boundary supported priors for other models and loss functions.