# zbMATH — the first resource for mathematics

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
Full Text:
##### References:
 [1] Berger, Estimation of quadratic functions (1991) [2] Cellier, Robust shrinkage estimators of the location parameter for elliptically symmetric distributions, J. Multivariate Anal. 29 pp 39– (1989) · Zbl 0678.62061 [3] Chow, A complete class theorem for estimating a noncentrality parameter, Ann. Statist. 15 pp 800– (1987) · Zbl 0627.62008 [4] Das Gupta, Power of the noncentral F-test: Effect of additional variates on Hotelling’s T2-test, J. Amer. Statist. Assoc. 69 pp 174– (1974) · Zbl 0285.62027 [5] Kubokawa, A unified approach to improving equivariant estimators (1991) · Zbl 0816.62021 [6] Kubokawa, Robust estimation of common coefficients under spherical symmetry, Ann. Inst. Statist. Math. 43 pp 677– (1991) · Zbl 0761.62037 [7] Kubokawa, On improved positive estimators of variance components (1991) [8] Neff, Further remarks on estimating the parameter of a noncentral chisquared distribution, Comm. Statist. A5 pp 65– (1976) · Zbl 0335.62020 [9] Perlman, Some remarks on estimating a noncentrality parameter, Comm. Statist. 4 pp 455– (1975) · Zbl 0302.62012 [10] Robert, On some accurate bounds for the quantiles of a noncentral chi-squared distribution, Statist. Probab. Lett. 10 pp 101– (1990) · Zbl 0699.62014 [11] Saxena, Estimation of the noncentrality parameter of a chi-squared distribution, Ann. Statist. 10 pp 1012– (1982)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.