Abstract
Models with commutative orthogonal block structure, COBS, have orthogonal block structure, OBS, and their least square estimators for estimable vectors are, as it will be shown, best linear unbiased estimator, BLUE. Commutative Jordan algebras will be used to study the algebraic structure of the models and to define special types of models for which explicit expressions for the estimation of variance components are obtained. Once normality is assumed, inference using pivot variables is quite straightforward. To illustrate this class of models we will present unbalanced examples before considering families of models. When the models in a family correspond to the treatments of a base design, the family is structured. It will be shown how, under quite general conditions, the action of the factors in the base design on estimable vectors, can be studied.
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First appeared in 1978 as a preprint (num. 159) of the Polish Academy of Science, Institute of Mathematics.
References
Caliński T, Kageyama S (2000) Block designs. A randomization approach. Volume I: analysis. Lecture notes in statistics. Springer, Berlin
Caliński T, Kageyama S (2003) Block designs. A randomization approach. Volume II: design. Lecture notes in statistics. Springer, Berlin
Carvalho F, Mexia JT, Oliveira MM (2008) Canonic inference and commutative orthogonal block structure. Discuss Math Prob Stat 28(2):171–181
Fonseca M, Mexia JT, Zmyślony R (2003) Estimators and tests for variance components in cross nested orthogonal models. Discuss Math Probab Stat 23:173–201
Fonseca M, Mexia JT, Zmyślony R (2006) Binary operation on Jordan algebras and orthogonal normal models. Linear Algebr Appl 417(1):75–86
Fonseca M, Mexia JT, Zmyślony R (2007) Jordan algebras, generating pivot variables and orthogonal normal models. J Interdiscip Math 10(2):305–326
Fonseca M, Mexia JT, Zmyślony R (2008) Inference in normal models with commutative orthogonal block structure. Acta et Commentationes Universitatis Tartunesis de Mathematica 12:3–16
Jordan P, von Neumann J, Wigner E (1934) On the algebraic generalization of the quantum mechanical formalism. Ann Math 35(1):29–64
Kariya T, Kurata H (2004) Generalized least squares. Wiley, New York
Khuri AI, Mathew T, Sinha BK (1998) Statistical tests for mixed linear models. Wiley series in probability and statistics. Wiley, New York
Lehmann EL, Casella G (2003) Theory of point estimation, 2nd edn. Springer Texts in Statistics
Mejza S (1992) On some aspects of general balance in designed experiments. Statistica 52(2):263–278
Mexia JT (1987) Multi-treatment regression designs. Trabalhos de Investigação, No 1. Departamento de Matemática, Faculdade de Ciências e Tecnologia, Universidade Nova de Lisboa
Mexia JT (1990) Best linear unbiased estimates, duality of \(F\) tests and the Scheff multiple comparison method in presence of controlled heterocedasticity. Comput Stat Data Anal 10:271–281
Michalski A, Zmyślony R (1996) Testing hypothesis for variance components in mixed linear models. Statistics 27(3–4):297–310
Moreira EE, Ribeiro AB, Mateus E, Mexia JT, Ottosen LM (2005a) Regressional modelling of electrodialytic removal of Cu, Cr and As from CCA timber waste: application to sawdust. Wood Sci Technol 39(4):291–309
Moreira EE, Mexia JT, Ribeiro AB, Mateus E, Ottosen LM (2005b) Regressional modelling of electrodialytic removal of Cu, Cr and As from CCA timber waste: application to wood chips. Listy Biometryczne 42(1):11–23
Moreira E, Mexia JT (2007) Multiple regression models with cross nested orthogonal base model. In: Proceedings of the 56th session of the ISI 2007—International Statistical Institute, Lisboa
Moreira E (2008) Família estruturada de modelos com base ortogonal: teoria e aplicações, Ph.D. Thesis, Faculdade de Ciências e Tecnologia, Universidade Nova de Lisboa (in Portuguese)
Nelder JA (1965a) The analysis of randomized experiments with orthogonal block structure. I. Block structure and the null analysis of variance. Proc R Soc Lond Ser A Math Phys Sci 283(1393):147–162
Nelder JA (1965b) The analysis of randomized experiments with orthogonal block structure. II. Treatment, structure and the general analysis of variance. Proc R Soc Lond Ser A Math Phys Sci 283(1393):163–178
Schott JR (1997) Matrix analysis for statistics. Wiley series in probability and statistics. Wiley, New York
Seely J (1970a) Linear spaces and unbiased estimators. Ann Math Stat 41(5):1725–1734
Seely J (1970b) Linear spaces and unbiased estimators. Application to a mixed linear model. Ann Math Stat 41(5):1735–1748
Seely J (1971) Quadratic subspaces and completeness. Ann Math Stat 42(2):710–721
Seely J, Zyskind G (1971) Linear spaces and minimum variance unbiased estimation. Ann Math Stat 42(2):691–703
Scheffé H (1959) The analysis of variance, Wiley series in probability and statistics. Wiley, New York
VanLeeuwen DM, Seely JF, Birkes DS (1998) Sufficient conditions for orthogonal designs in mixed linear models. J Stat Plan Inference 73(1–2):373–389
VanLeeuwen DM, Birkes DS, Seely JF (1999) Balance and orthogonality in designs for mixed classification models. Ann Stat 27(6):1927–1947
Zmyślony R (1980) A characterization of Best Linear Unbiased Estimators in the general linear model. Lecture Notes in Statistics, vol 2. Springer, pp 365–373
Zmyślony R, Drygas H (1992) Jordan algebras and bayesian quadratic estimation of variance components. Linear Algebra Appl 168:259–275
Acknowledgments
The authors would like to thank the anonymous referees for useful comments and suggestions. This work was partially supported by CMA / FCT / UNL, under the project PEst-OE/MAT/UI0297/2014, and by the Center of Mathematics, University of Beira Interior under the project PEst-OE/MAT/UI0212/2014.
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Carvalho, F., Mexia, J.T., Santos, C. et al. Inference for types and structured families of commutative orthogonal block structures. Metrika 78, 337–372 (2015). https://doi.org/10.1007/s00184-014-0506-8
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DOI: https://doi.org/10.1007/s00184-014-0506-8