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By Shapley H.

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Further, Marchand and MacGibbon (2000, p. 2. 23). 15) are monotone, so there is no contradiction. 3. 2) forgot to mention a necessary condition on (m, ε), namely, m + ε ≤ 1/2. More about dominators when X ∼ Bin(n, θ) with m ≤ θ ≤ 1 − m for a known m ∈ (0, 1/2) can be found in Perron (2003). He uses squared-error loss and gives sufficient conditions for an estimator to dominate the MLE. For Moors’ (1985) dominator of the MLE Perron shows that it is the Bayes estimator with √respect to a symmetric prior √ om {m, 1 − m} if and only if 1 − 2m is ≤ 1/ n when n is odd and ≤ 1/( n − 1) when n is even.

They define restricted Stein–rule MLEs θˆRS and positive restricted Stein–rule MLEs θˆP RS of θ for θ ∈ Θk+1 . One of their results states that θˆP RS dominates θˆRS which dominates the (restricted) MLE. On the relationship of their work with that of Chang (1981, 1982) Sengupta and Sen say that the full impact of shrinkage has not been incorporated in the estimators considered by Chang and they give dominators of Chang’s (1982) dominators. 3) weaken the conditions on Θ for one of Sengupta and Sen’s (1991) domination results and Ouassou and Strawderman (2002) and Fourdrinier, Ouassou and Strawderman (2003) generalize and extend the Chang (1981, 1982) and Sengupta and Sen (1991) results to spherically symmetric distributions.

K are independent and the Xi,j , j = 1, . . , ni have Lebesgue density f (x/θi )/θi , i = 1, . . , k. Using scalek invariant squared-error loss, the Pitman estimator of θ = (θ1 , . . , θk ) ∈ R+ is, when it has a finite risk function, the minimum-risk-equivariant estimator and also the Bayes estimator for the uniform prior for log θ on (−∞, ∞)k . The i-th element, i = 1, . . , k, of this Pitman estimator is the Pitman estimator of θi based on Xi,1 , . . , Lehmann, 1983, p. 177; or Lehmann and Casella, 1998, p.

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A Contribution to the Study of Galactic Dimensions by Shapley H.


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