James-Stein type estimators for ordered normal means

摘要

Suppose independent observations X_1, X_2, ..., X_k are available from k ( ≥ 2) normal populations having means θ_1, θ_2, ..., θ_k, respectively, and common variance unity. The means are known to be ordered, that is, θ_1 ≤ θ_2 ≤ • • • ≤ θ_k. Two commonly used estimators for simultaneous estimation of θ = (θ_1, θ_2,..., θ_k) are δ_(MLE) the order restricted maximum likelihood estimator (MLE), and δ_p, the generalized Bayes estimator of θ with respect to the uniform prior on the restricted space Ω = { θ∈ R~k: θ_1 ≤ θ_2 ≤ • • • ≤ θ_k}, where R~k denotes the k-dimensional Euclidean space. Both δ_(MLE) and δ_p improve the usual unrestricted MLE X= (X_1, X_2,... , X_k) and are minimax. But δ_(MLE) is inadmissible for k ≥ 2 and δ_p is inadmissible for k ≥ 3. However, no dominating estimators are yet known. Using Brown''s [Brown, L.D., 1979, A heuristic method for determining admissibility of estimators-with applications. Annals of Statistics, 7, 960-994] heuristic approach for proving admissibility or inadmissibility of estimators, we propose some classes of James-Stein type estimators and show, through a simulation study, that many of these estimators dominate δ_p and 腳(MLE).