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Estimation of noncentrality parameters. (English) Zbl 0787.62023
Since the naive improvement \(\delta^ +_ 0X=\max\{X-p,0\}\) (of the unbiased estimator \(\delta^ +_ 0(X)=X-p\) of the noncentrality parameter \(\Lambda\) of the \(\chi^ 2\)-distribution) is under the quadratic loss \((\delta-\Lambda)^ 2\) inadmissible, the authors proposed the estimator \(\delta(\varphi)=X-\varphi(X)\), which is a generalization of the earlier used estimator \(\delta^{PR}(X)=X-p+a/X\).
Conditions (in fact the bounds) for \(\varphi(X)\) are given under which \(\delta(\varphi)\) dominates \(\delta_ 0(X)\), and results of a Monte Carlo study are presented (for \(\delta_ 0,\delta^ +_ 0\) and for special choices of \(\varphi)\).
A similar improvement is achieved (in the framework of linear regression models) for the estimator of the value \(\beta^ TX^ TX\beta/\sigma^ 2\) under the assumption that the fluctuations in the model follow some elliptically contoured distribution (e.c.d.). Two possibilities are considered: (i) the type of the e.c.d. is known, (ii) the type is not specified. The result in the case (ii) represents a robustified modification of the result for the case (i), and the method is a version of the Stein effect.

MSC:
62F10 Point estimation
62C15 Admissibility in statistical decision theory
62A01 Foundations and philosophical topics in statistics
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