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Bayes, admissible, and minimax linear estimators in linear models with restricted parameter space. (English) Zbl 0686.62019
Summary: We consider a linear model E Y\(=X\alpha\), where the p-dimensional parameter vector \(\alpha\) is restricted by \(A\alpha\in \Omega\). The matrix A is (q\(\times p)\) of rank q, and \(\Omega\) is a compact subset of \({\mathbb{R}}^ q\), which contains zero and is symmetric to zero. The dispersion matrix of Y is known and positive definite. For estimation of a given s-dimensional linear parametric function, we confine to the class of all linear estimators with bounded mean squared error functions. The Bayes estimators within this class are found, and they are shown to coincide with the admissible estimators within this class. The minimax estimator is characterized as the linear Bayes estimator against a least favourable prior distribution, and a result helpful to find a least favourable prior is proved.
Applications are given firstly to the case of full parameter restrictions \((q=p)\), and some new results on linear admissible and linear minimax estimation are obtained. Secondly we consider the case of k unrestricted and p-k restricted parameters and estimation of unrestricted parameters. In particular the question is answered, when the BLUE from the simplified model, which ignores the restricted parameters, is the linear minimax estimator under the present model.

MSC:
62F15 Bayesian inference
62J05 Linear regression; mixed models
62H12 Estimation in multivariate analysis
62F10 Point estimation
62C15 Admissibility in statistical decision theory
62C20 Minimax procedures in statistical decision theory
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