Variance components estimation in generalized orthogonal models

ArticleOriginal scientific text

Title

Authors ¹, ¹, ², ³

Mathematics Scientific Area Instituto Superior de Engenharia de Lisboa, Rua Conselheiro Emídio Navarro, 01, 1959-007 Lisboa, Portugal
Mathematics Department, Universidade da Beira Interior, 6201-001 Covilhã, Portugal
Department of Mathematics, Faculty of Science and Technology, New University of Lisbon, Monte de Caparica 2829-516 Caparica, Portugal

The model

y = \sum_{j = 1}^{w} X_{j} β ̲_{j} + e ̲

is generalized orthogonal if the orthogonal projection matrices on the range spaces of matrices

X_{j}

, j = 1, ..., w, commute. Unbiased estimators are obtained for the variance components of such models with cross-nesting.

generalized orthogonal models, variance components, generalized cross nesting designs

C. Fernandes, P. Ramos and J.T. Mexia, Optimização de Delineamentos em Escada Simples, Actas do XII Congresso Anual da SPE (2005), 247-253.
C. Fernandes, P. Ramos and J.T. Mexia, Optimization of Nested Step Designs, Biometrical Letters 42-2 (2005), 143-151.
C. Fernandes, P. Ramos and J.T. Mexia, Evenness conditions for four cross nested models, Biometrical Letters 43-2 (2006), 109-130.
M. Fonseca, J.T. Mexia and R. Zmyślony, Exact distributions for the generalized F tests, Discussiones Mathematicae - Probability and Statistics 22 (2002), 37-51.
E.L. Lehmann, Testing Statistical Hypotesis, Jonh Willey and Sons, New York 1952.
J. Malley, Statistical Applications of Jordan Algebras, Springer-Verlag 1994.
A. Michalski and R. Zmyślony, Testing hypothesis for variance components in mixed linear models, Statistics 27 (1996), 297-310.
A. Michalski and R. Zmyślony, Testing hypothesis for linear functions of parameters in mixed linear models, Tatra Mountain Mathematical Publications 1999.
J. Nelder, The interpretation of negative components of variance, Biometrika 41 (1954), 544-548.