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Some applications of Anderson’s inequality to some optimality criteria of the generalized least squares estimator. (English) Zbl 0753.62017

Summary: The article shows that the generalized least squares estimator \(\hat\beta\) of the regression coefficient \(\beta\) is a generalized Bayes estimator for a family of elliptically symmetric unimodal distributions and for a large class of loss functions which are nondecreasing in the generalized Euclidean distance between the parameter and the estimator. For a group of location-scale transformations, \(\hat\beta\) is shown to be the Pitman closest and to dominate universally in the class of equivariant estimators of \(\beta\). Also \(\hat\beta\) is shown to be a minimax estimator of \(\beta\). Furthermore, it can be shown that \(\hat\beta\) is the posterior Pitman closest. In proving all these results, T. W. Anderson’s inequality [Proc. Amer. Math. Soc. 6, 170-176 (1955; Zbl 0066.374)] has been used as a basic tool.

MSC:

62F15 Bayesian inference
62J05 Linear regression; mixed models
62A01 Foundations and philosophical topics in statistics
62C20 Minimax procedures in statistical decision theory
62H12 Estimation in multivariate analysis

Citations:

Zbl 0066.374
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References:

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