Geodesic
Modified minimax quadratic estimation of variance components
Kybernetika, Tome 34 (1998) no. 5, pp. 535-543 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

Voir la notice de l'article

The paper deals with modified minimax quadratic estimation of variance and covariance components under full ellipsoidal restrictions. Based on the, so called, linear approach to estimation variance components, i. e. considering useful local transformation of the original model, we can directly adopt the results from the linear theory. Under normality assumption we can can derive the explicit form of the estimator which is formally find to be the Kuks–Olman type estimator.
The paper deals with modified minimax quadratic estimation of variance and covariance components under full ellipsoidal restrictions. Based on the, so called, linear approach to estimation variance components, i. e. considering useful local transformation of the original model, we can directly adopt the results from the linear theory. Under normality assumption we can can derive the explicit form of the estimator which is formally find to be the Kuks–Olman type estimator.
Classification : 62C20, 62F10, 62F30, 62F35, 62J10
Keywords: ellipsoidal restrictions 
@article{KYB_1998_34_5_a2,
     author = {Witkovsk\'y, Viktor},
     title = {Modified minimax quadratic estimation of variance components},
     journal = {Kybernetika},
     pages = {535--543},
     year = {1998},
     volume = {34},
     number = {5},
     mrnumber = {1663728},
     zbl = {1274.62477},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/KYB_1998_34_5_a2/}
}
TY  - JOUR
AU  - Witkovský, Viktor
TI  - Modified minimax quadratic estimation of variance components
JO  - Kybernetika
PY  - 1998
SP  - 535
EP  - 543
VL  - 34
IS  - 5
UR  - http://geodesic.mathdoc.fr/item/KYB_1998_34_5_a2/
LA  - en
ID  - KYB_1998_34_5_a2
ER  - 
%0 Journal Article
%A Witkovský, Viktor
%T Modified minimax quadratic estimation of variance components
%J Kybernetika
%D 1998
%P 535-543
%V 34
%N 5
%U http://geodesic.mathdoc.fr/item/KYB_1998_34_5_a2/
%G en
%F KYB_1998_34_5_a2
Witkovský, Viktor. Modified minimax quadratic estimation of variance components. Kybernetika, Tome 34 (1998) no. 5, pp. 535-543. http://geodesic.mathdoc.fr/item/KYB_1998_34_5_a2/

[1] Gaffke N., Heiligers B.: Bayes, admissible, and minimax linear estimators in linear models with restricted parameter space. Statistics 20 (1989), 4, 487–508 | DOI | MR | Zbl

[2] Heiligers B.: Linear Bayes and minimax estimation in linear models with partially restricted parameter space. J. Statist. Plann. Inference 36 (1993), 175–184 | DOI | MR | Zbl

[3] Kozák J.: Modified minimax estimation of regression coefficients. Statistics 16 (1985), 363–371 | DOI | MR | Zbl

[4] Kubáček L., Kubáčková L., Volaufová J.: Statistical Models with Linear Structures. Publishing House of the Slovak Academy of Sciences, Bratislava 1995

[5] Pilz J.: Minimax linear regression estimation with symmetric parameter restrictions. J. Statist. Plann. Inference 13 (1986), 297–318 | DOI | MR | Zbl

[6] Pukelsheim F.: Estimating variance components in linear models. J. Multivariate Anal. 6 (1976), 626–629 | DOI | MR | Zbl

[7] Rao C. R.: Estimation of variance and covariance components – MINQUE theory. J. Multivariate Anal. 1 (1971), 257–275 | DOI | MR | Zbl

[8] Rao C. R.: Minimum variance quadratic unbiased estimation of variance components. J. Multivariate Anal. 1 (1971), 445–456 | DOI | MR | Zbl

[9] Rao C. R.: Unified theory of linear estimation. Sankhyā Ser. B 33 (1971), 371–394 | MR | Zbl

[10] Rao C. R., Kleffe J.: Estimation of Variance Components and Applications. Statistics and Probability, Volume 3. North–Holland, Amsterdam – New York – Oxford – Tokyo 1988 | MR | Zbl

[11] Rao C. R., Mitra K.: Generalized Inverse of Matrices and Its Applications. Wiley, New York – London – Sydney – Toronto 1971 | MR | Zbl

[12] Searle S. R., Casella, G., McCulloch, Ch. E.: Variance Components. (Wiley Series in Probability and Mathematical Statistics.) Wiley, New York – Chichester – Brisbane – Toronto – Singapore 1992 | MR | Zbl

[13] Volaufová J.: A brief survey on the linear methods in variance-covariance components model. In: Model–Oriented Data Analysis (W. G. Müller, H. P. Wynn, and A. A. Zhigljavsky, eds.), Physica–Verlag, Heidelberg 1993, pp. 185–196 | MR

[14] Volaufová J., Witkovský V.: Estimation of variance components in mixed linear model. Appl. Math. 37 (1992), 139–148

[15] Zyskind G.: On canonical forms, nonnegative covariance matrices and best and simple least square estimator in linear models. Ann. Math. Statist. 38 (1967), 1092–1110 | DOI | MR