[Article] Studies of Magnitudes in Star Clusters V. Further by Shapley H.

By Shapley H.

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Extra info for [Article] Studies of Magnitudes in Star Clusters V. Further Evidence of the Absence of Scattering of Light in Space

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K,1 , θk,2 ) satisfies θi,1 ≤ θi,2 , i = 1, . . , k. He estimates θ under squared-error loss. 4). Chang (1991) shows that, analogous to Stein’s (1956) result, the vector of these Pitman estimators is inadmissible for estimating θ when k ≥ 2. One of his two classes of James–Stein-type dominators is given by (1 − c/S)δP (X), where δP (X) = (δP,1 (X1 ), . . , δP,k (Xk )) is the vector of Pitman estimators, k 2 S = i=1 (δP,i (Xi )) and 0 < c ≤ 4(k − 1). He has a second class of dominators for the case where k ≥ 3.

Chang (1991) shows that, analogous to Stein’s (1956) result, the vector of these Pitman estimators is inadmissible for estimating θ when k ≥ 2. One of his two classes of James–Stein-type dominators is given by (1 − c/S)δP (X), where δP (X) = (δP,1 (X1 ), . . , δP,k (Xk )) is the vector of Pitman estimators, k 2 S = i=1 (δP,i (Xi )) and 0 < c ≤ 4(k − 1). He has a second class of dominators for the case where k ≥ 3. Chang (1982) looks at the case where Θ = {θ | θi ≥ 0, i = 1, . . , k} with squared error loss and X ∼ Nk (θ, I).

3)) a least favourable prior on Θ with finite support and a minimax estimator which is Bayes with respect to this prior. The number of points in this support increases with the “size” of Θ. The problem of finding these points and the prior probabilities can only seldom be solved analytically. As will be seen below, when k = 1 and θ = [m1 , m2 ], analytical results have been obtained for small values of m2 − m1 or (m2 /m1 ) − 1 for particular cases. For other particular cases, numerical results are available.

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