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Constrained minimax estimation of the mean of the normal distribution with known variance. (English) Zbl 0745.62021

Summary: We discuss the estimation of the mean of a normal distribution with variance 1. The main question in this work is the existence and computation of a least favorable distribution among all the prior distributions satisfying a given set of constraints.
We show that if this distribution is bounded from above on some even moment, then the least favorable distribution exists and it is either normal or discrete. The support of the discrete distribution function does not have any accumulation point. The least favorable distribution is normal if and only if the second moment is bounded from above, without any other relevant constraint. These theorems shed light on the James- Stein estimator as the minimax estimator for a prior with unknown variance.

MSC:

62F10 Point estimation
62C20 Minimax procedures in statistical decision theory
62F15 Bayesian inference
62F30 Parametric inference under constraints
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