Estimation and orthogonal block structure

Year 2016, Volume: 45 Issue: 2, 541 - 548, 01.04.2016

Sandra S. Ferreirax Célia Nunes Dário Ferreira Elsa Moreira João Tiago Mexia

https://izlik.org/JA86AN43RC

Abstract

Estimators with good behaviors for estimable vectors and variance components are obtained for a class of models that contains the well known models with orthogonal block structure, OBS, see [15], [16] and [1], [2]. The study observations of these estimators uses commutative Jordan Algebras, CJA, and extends the one given for a more restricted class of models, the models with commutative orthogonal block structure, COBS, in which the orthogonal projection matrix on the space spanned by the means vector commute with all variance-covariance matrices, see [7].

Keywords

BLUE , LSE , OBS , UMVUE , Variance components

References

• Caliński, T., Kageyama, S. Block Designs: A Randomization Approach, Volume I: Analysis, Lect. Notes Stat. 150, Springer, New York, 2000.
• Caliński, T., Kageyama, S. Block Designs: A Randomization Approach, Volume II: Design, Lect. Notes Stat., 170, Springer, New York, 2003.
• Carvalho, F., Mexia, J.T., Oliveira, M. Canonic inference and commutative orthogonal block structure. Discuss. Math. Probab. Stat., 28, 171-181, 2008.
• Ferreira, S.S., Ferreira, D., Fernandes, C., Mexia, J.T. Orthogonal Mixed Models and Perfect Families of Symmetric Matrices, Book of Abstracts, 56-th Session of the International Statistical Institute, ISI, Lisboa, 2007.
• Fonseca, M., Mexia, J.T., Zmyślony, R. Binary operations on Jordan algebras and orthogonal normal models, Linear Algebra Appl., 417, 75-86, 2006. 548
• Fonseca, M., Mexia, J.T., Zmyślony, R. Jordan algebras, generating pivot variables and orthogonal normal models, J. Interdiscip. Math., 10(2), 305-326, 2007.
• Fonseca, M., Mexia, J.T., Zmyślony, R. Inference in normal models with commutative orthogonal block structure. Acta Comment. Univ. Tartu. Math., 12, 3-16, 2008.
• Fonseca, M., Mexia, J.T., Zmyślony, R. Least squares and generalized least squares in models with orthogonal block structure, J. Statist. Plann. Inference, 140(5), 1346-1352, 2010.
• Henderson, C. R. Statistical Methods in Animal Improvement: Historical Overview. In Gianola, Daniel and Hammond, Keith. Advances in Statistical Methods for Genetic Improvement of Livestock. Springer-Verlag Inc., 2-14, 1990.
• Houtman, A.M., Speed, T.P. Balance in designed experiments with orthogonal block structure. Ann. Statist., 11, 1069-1085, 1983.
• Jesus, V., Mexia, J.T., Zmyślony, R. Binary operations and canonical forms for factorial and related models, Linear Algebra Appl., 430(10) 2781-2797, 2009.
• Jordan, P., Neumann, J. von, Wigner, E. On an Algebraic Generalization of the Quantum Mechanical Formalism. Ann. of Math. (Princeton), 35(1), 29-64, 1934.
• Mexia, J.T. Best linear unbiased estimates, duality of F tests and the Scheffé multiple comparison method in the presence of controlled heteroscedasticity. Comput. Stat. Data An., 10(3) 271-281, 1990.
• Mexia, J.T., Vaquinhas, R., Fonseca, M., Zmyślony, R. COBS: Segregation, Matching, Crossing and Nesting. Latest Trends on Applied Mathematics, Simulation, Modeling, 4th International Conference on Applied Mathematics, Simulation, Modelling (ASM’10) 249-255, 2010.
• Nelder, J.A. The analysis of randomized experiments with orthogonal block structure. I. Block structure and the null analysis of variance. Proceedings of the Royal Society (London), Series A, 273, 147-162, 1965a.
• Nelder, J.A. The analysis of randomized experiments with orthogonal block structure. II. Treatment structure and the general analysis of variance. Proceedings of the Royal Society (London), Series A, 273, 163-178, 1965b.
• Schott, J.R. Matrix Analysis for Statistics. John Wiley & Sons, New York, 1997.
• Searle, S.R., Casella, G., McCulloch, C.E. Variance Components. John Wiley & Sons, New York, 1992.
• Seely, J. Quadratic subspaces and completeness. Ann. Math. Stat., 42(2), 710-721, 1971.
• Silvey, S.D. Statistical Inference. Reprinted. Chapman & Hall, 1975.
• VanLeeuwen, D.M., Seely, J.F., Birkes, D.S. Sufficient conditions for orthogonal designs in mixed linear models. J. Statist. Plann. Inference, 73, 373-389, 1998.
• Zyskind, G. . On canonical forms, non-negative covariance matrices and best simple linear least squares estimators in linear models. Ann. Math. Stat., 38, 1092-1109, 1967.
• Zmyślony, R. A characterization of best linear unbiased estimators in the general linear model, Mathematical Statistics and Probability Theory, Proceedings Sixth International Conference, Wisla, Lecture Notes in Statistics, Springer, New York-Berlin, 2, 365-373, 1978.