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.
Commutative orthogonal block structure Commutative Jordan algebras Estimation Mixed linear models
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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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