×

zbMATH — the first resource for mathematics

Modified minimax quadratic estimation of variance components. (English) Zbl 1274.62477
Summary: 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 of variance components, i.e., considering useful local transformations of the original model, we can directly adopt the results from the linear theory. Under a normality assumption we can can derive the explicit form of the estimator which is formally find to be a Kuks-Olman type estimator.
MSC:
62J10 Analysis of variance and covariance (ANOVA)
62C20 Minimax procedures in statistical decision theory
62F30 Parametric inference under constraints
PDF BibTeX XML Cite
Full Text: Link
References:
[1] Gaffke N., Heiligers B.: Bayes, admissible, and minimax linear estimators in linear models with restricted parameter space. Statistics 20 (1989), 4, 487-508 · Zbl 0686.62019 · doi:10.1080/02331888908802199
[2] Heiligers B.: Linear Bayes and minimax estimation in linear models with partially restricted parameter space. J. Statist. Plann. Inference 36 (1993), 175-184 · Zbl 0780.62027 · doi:10.1016/0378-3758(93)90122-M
[3] Kozák J.: Modified minimax estimation of regression coefficients. Statistics 16 (1985), 363-371 · Zbl 0588.62108 · doi:10.1080/02331888508801866
[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 · Zbl 0602.62054 · doi:10.1016/0378-3758(86)90141-2
[6] Pukelsheim F.: Estimating variance components in linear models. J. Multivariate Anal. 6 (1976), 626-629 · Zbl 0355.62061 · doi:10.1016/0047-259X(76)90010-5
[7] Rao C. R.: Estimation of variance and covariance components - MINQUE theory. J. Multivariate Anal. 1 (1971), 257-275 · Zbl 0223.62086 · doi:10.1016/0047-259X(71)90001-7
[8] Rao C. R.: Minimum variance quadratic unbiased estimation of variance components. J. Multivariate Anal. 1 (1971), 445-456 · Zbl 0259.62061 · doi:10.1016/0047-259X(71)90019-4
[9] Rao C. R.: Unified theory of linear estimation. Sankhyā Ser. B 33 (1971), 371-394 · Zbl 0236.62048
[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 · Zbl 0645.62073
[11] Rao C. R., Mitra K.: Generalized Inverse of Matrices and Its Applications. Wiley, New York - London - Sydney - Toronto 1971 · Zbl 0261.62051
[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 · Zbl 1108.62064
[13] Volaufová J.: A brief survey on the linear methods in variance-covariance components model. Model-Oriented Data Analysis (W. G. Müller, H. P. Wynn, and A. A. Zhigljavsky, Physica-Verlag, Heidelberg 1993, pp. 185-196 · Zbl 0875.62320
[14] Volaufová J., Witkovský V.: Estimation of variance components in mixed linear model. Appl. Math. 37 (1992), 139-148 · Zbl 0746.62066 · eudml:15705
[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 · Zbl 0171.17103 · doi:10.1214/aoms/1177698779
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.