We study the algebraic structure of orthogonal models thus of mixed models whose variance covariance matrices are all positive semi definite, linear combinations of known pairwise orthogonal projection matrices, POOPM, and whose least square estimators, LSE, of estimable vectors are best linear unbiased estimator, BLUE, whatever the variance components, so they are uniformly BLUE, UBLUE. From the results of the algebraic structure we will get explicit expression for the LSE of these models.
@article{bwmeta1.element.bwnjournal-article-doi-10_7151_dmps_1176,
author = {Artur Pereira and Miguel Fonseca and Jo\~ao Tiago Mexia},
title = {Orthogonal models: Algebraic structure and explicit estimators for estimable vectors},
journal = {Discussiones Mathematicae Probability and Statistics},
volume = {35},
year = {2015},
pages = {29-44},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_7151_dmps_1176}
}
Artur Pereira; Miguel Fonseca; João Tiago Mexia. Orthogonal models: Algebraic structure and explicit estimators for estimable vectors. Discussiones Mathematicae Probability and Statistics, Tome 35 (2015) pp. 29-44. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_7151_dmps_1176/
[000] [1] A. Areia and F. Carvalho, Perfect families: an appliacation to orthogonal and error orthogonal models, AIP Conf. Proc. Numerical Analysis and Applied Mathematics: International Conferences of Numerical Analysis and Applied Mathematics 1558 (2013), 841-846.
[001] [2] T. Calinski and S. Kageyama, Block Designs: A Randomization Approach. Vol. I: Analysis , Lecture Notes in Statistics, Springer, 2000, 150.
[002] [3] T. Calinski and S. Kageyama, Block Designs: A Randomization Approach. Vol. II: Design, Lecture Notes in Statistics, Springer, 2003, 170.
[003] [4] F. Carvalho, J.T. Mexia and M.M. Oliveira, Canonic inference and commutative orthogonal block structure, Discuss. Math. Probab. and Stat. 28 (2) (2008), 171-181. | Zbl 1208.62093
[004] [5] F. Carvalho, J.T. Mexia and C. Santos, Commutative orthogonal block structure and error orthogonal models, Electronic Journal of Linear Algebra 25 (2013), 119-128. | Zbl 1283.15087
[005] [6] S.S. Ferreira, D. Ferreira, C. Fernandesa and J.T. Mexia, Orthogonal models and perfect families of symmetric matrices, Bulletin of the International Statistical Institute. Proc. ISI 2007, Lisboa 22-28 August (2007), 3252-3254.
[006] [7] M. Fonseca, J.T. Mexia and R. Zmyślony, Inference in normal models with commutative orthogonal block structure, Acta et Commentationes Universitatis Tartunesis de Mathematica (2008), 3-16. | Zbl 1166.62313
[007] [8] A. Houtman and T.P. Speed, Balance in designed experiments with orthogonal block structure, The annals of Statistics 11 (4) (1983), 1069-1085. | Zbl 0566.62065
[008] [9] E.L. Lehmann and G. Casela, Theory of Point estimation, 2nd ed. (Springer Tests Statistics, New York, Springer, 1998).
[009] [10] S. Mejza, On some aspects of general balance in designed experiments, Statistica 52 (1992), 263-278. | Zbl 0770.62060
[010] [11] J.T. Mexia, R. Vaquinhas, M. Fonseca and R. Zmyślony, COBS: segregation, matching, crossing and nesting, Latest Trends on Applied Mathematics, Simulation, Modelling (2010), 249-255.
[011] [12] J.A. Nelder, The analysis of randomized experiments with orthogonal block structure and the null analysis of variance, Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences 283 (1393), 147-162. | Zbl 0124.10703
[012] [13] J.A. Nelder, The analysis of randomized experiments with orthogonal block structure II. Treatment structure and the general analysis of variance, Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences 283 (1393), 163-178. | Zbl 0124.10703
[013] [14] J.R. Schott, Matrix Analysis for Statistics (Wiley Series in Probability and Statistics, 1997). | Zbl 0872.15002
[014] [15] J. Seely, Quadratic subspaces and completeness, The Annals of Mathematical Statistics 42 (1971), 710-721. | Zbl 0249.62067
[015] [16] D.M. Vanleeuwen, D.S. Birks, J.F. Seely, J. Mills, J.A. Greenwood and C.W. Jones, Sufficient conditions for orthogonal designs in mixed linear models, Journal of Statistics Planning and Inference 73 (1998), 373-389.
[016] [17] R. Zmyślony, A characterization of the best linear unbiased estimators in the general linear model, Lecture Notes Statistics 2 (1978), 365-373.