×

Best unbiased estimates for parameters of three-level multivariate data with doubly exchangeable covariance structure. (English) Zbl 1376.62035

Summary: The article addresses the best unbiased estimators of doubly exchangeable covariance structure, an extension of block exchangeable covariance structure, for three-level multivariate data. Under multivariate normality, the free-coordinate approach is used to obtain linear and quadratic estimates for the model parameters that are unbiased, sufficient, complete and consistent. Data from a clinical trial study is analyzed to illustrate the application of the obtained results.

MSC:

62H12 Estimation in multivariate analysis
62J10 Analysis of variance and covariance (ANOVA)
62P10 Applications of statistics to biology and medical sciences; meta analysis
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Coelho, C. A.; Roy, A., Testing the hypothesis of a doubly exchangeable covariance matrix for elliptically contoured distributions, (2014), College of Business, The University of Texas at San Antonio, Working Paper No. 0002MSS-253-2014
[2] Fonseca, M.; Mexia, J. T.; Zmyślony, R., Least squares and generalized least squares in models with orthogonal block structure, J. Statist. Plann. Inference, 140, 1346-1352, (2010) · Zbl 1181.62083
[3] Drygas, H., The coordinate-free approach to Gauss-Markov estimation, (1970), Springer Berlin, Heidelberg · Zbl 0215.26504
[4] Gnot, S.; Klonecki, W.; Zmyślony, R., Uniformly minimum variance unbiased estimation in Euclidean vector spaces, Bull. de l. Academie Polon. des Sciences XXIV, vol. 4, 281-286, (1976) · Zbl 0345.62020
[5] Gnot, S.; Klonecki, W.; Zmyślony, R., Uniformly minimum variance unbiased estimation in various classes of estimators, Math. Oper.forsch. Stat., Ser. Stat., 8, 199-210, (1977) · Zbl 0386.62050
[6] Gnot, S.; Klonecki, W.; Zmyślony, R., Best unbiased estimation: a coordinate free-approach, Probab. Stat., 1, 1-13, (1980) · Zbl 0526.62061
[7] von Jordan, P.; Neumann, J.; Wigner, E., On an algebraic generalization of the quantum mechanical formalism, Ann. of Math., 35, 29-64, (1934) · JFM 60.0902.02
[8] A. Kozioł, A. Roy, R. Zmyślony, R. Leiva, M. Fonseca, Best unbiased estimators for doubly multivariate data, submitted for publication, 2015.
[9] Kruskal, W., When are Gauss-Markov and least squares estimators identical? A coordinate-free approach, Ann. Math. Stat., 39, 70-75, (1968) · Zbl 0162.21902
[10] Leiva, R.; Roy, A., Classification rules for triply multivariate data with an ar(1) correlation structure on the repeated measures over time, J. Statist. Plann. Inference, 139, 2598-2613, (2009) · Zbl 1162.62059
[11] Leiva, R.; Roy, A., Linear discrimination for multi-level multivariate data with separable means and jointly equicorrelated covariance structure, J. Statist. Plann. Inference, 141, 1910-1924, (2011) · Zbl 1207.62137
[12] Leiva, R.; Roy, A., Linear discrimination for three-level multivariate data with separable additive mean vector and doubly exchangeable covariance structure, Comput. Statist. Data Anal., 56, 1644-1661, (2012) · Zbl 1243.62094
[13] Roy, A., A two-stage principal component analysis of symbolic data using equicorrelated and jointly equicorrelated covariance structures, (2014), College of Business, The University of Texas at San Antonio, Working Paper No. 0006MSS-253-2014
[14] Roy, A.; Leiva, R., Discrimination with jointly equicorrelated multi-level multivariate data, Adv. Data Anal. Classif., 1, 175-199, (2007) · Zbl 1182.62142
[15] Roy, A.; Leiva, R., Estimating and testing a structured covariance matrix for three-level multivariate data, Comm. Statist. Theory Methods, 40, 1945-1963, (2011) · Zbl 1216.62095
[16] Roy, A.; Fonseca, M., Linear models with doubly exchangeable distributed errors, Comm. Statist. Theory Methods, 41, 2545-2569, (2012) · Zbl 1270.62102
[17] Roy, A.; Zmyślony, R.; Fonseca, M.; Leiva, R., Optimal estimation for doubly multivariate data in blocked compound symmetric covariance structure, J. Multivariate Anal., 144, 81-90, (2016) · Zbl 1328.62343
[18] Seely, J. F., Quadratic subspaces and completeness, Ann. Math. Stat., 42, 710-721, (1971) · Zbl 0249.62067
[19] Seely, J. F., Minimal sufficient statistics and completeness for multivariate normal families, Sankhya, Ser. A, 39, 170-185, (1977) · Zbl 0409.62004
[20] Zmyślony, R., On estimation of parameters in linear models, Appl. Math. (Warsaw) XV, 3, 271-276, (1976) · Zbl 0401.62049
[21] Zmyślony, R., A characterization of best linear unbiased estimators in the general linear model, Lecture Notes in Statistics, vol. 2, 365-373, (1978)
[22] Zmyślony, R., Completeness for a family of normal distributions, Mathematical Statistics, vol. 6, 355-357, (1980), Banach Center Publications
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.