James–Stein type estimators for ordered normal means

Abstract

Suppose independent observations X ₁, X ₂, …, X _k are available from k (≥2) normal populations having means θ₁, θ₂, …, θ _k , respectively, and common variance unity. The means are known to be ordered, that is, θ₁≤θ₂≤···≤θ _k . Two commonly used estimators for simultaneous estimation of θ=(θ₁, θ₂, …, 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 : θ₁≤θ₂≤···≤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 δ _MLE.

Keywords:

• Ordered parameters
• Maximum likelihood estimator
• Pitman estimator
• Generalized Bayes estimator
• Mixed estimators
• James–Stein estimator
• Translation invariance

Acknowledgements

The authors are thankful to a referee for some very useful suggestions which have substantially improved the paper.