Jordan algebras, generating pivot variables and orthogonal normal models

Fonseca, Miguel; Mexia, João T.; Zmyślony, Roman

doi:10.1080/09720502.2007.10700493

Cited by 16 publications

(13 citation statements)

References 12 publications

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“…For an example satisfying Assumption 5 but not Assumption 6, use the five factors in a 2 × 2 Latin square: see Tjur [42, §7.3]. [17,18,19] call the matrices XX and M 0 , . .…”

Section: Assumption 1 [Linear Expectation]mentioning

confidence: 99%

Estimability of variance components when all model matrices commute

Bailey

Ferreira

Nunes

2016

Linear Algebra and its Applications

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This paper deals with estimability of variance components in mixed models when all model matrices commute. In this situation, it is well known that the best linear unbiased estimators of fixed effects are the ordinary least squares estimators. If, in addition, the family of possible variance-covariance matrices forms an orthogonal block structure, then there are the same number of variance components as strata, and the variance components are all estimable if and only if there are non-zero residual degrees of freedom in each stratum.We investigate the case where the family of possible variance-covariance matrices, while still commutative, no longer forms an orthogonal block structure. Now the variance components may or may not all be estimable, but there is no clear link with residual degrees of freedom. Whether or not they are all estimable, there may or may not be uniformly best unbiased quadratic estimators of those that are estimable. Examples are given to demonstrate all four possibilities.

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“…For an example satisfying Assumption 5 but not Assumption 6, use the five factors in a 2 × 2 Latin square: see Tjur [42, §7.3]. [17,18,19] call the matrices XX and M 0 , . .…”

Section: Assumption 1 [Linear Expectation]mentioning

confidence: 99%

Estimability of variance components when all model matrices commute

Bailey

Ferreira

Nunes

2016

Linear Algebra and its Applications

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“…Now, let B r be the r-algebra of the borelian sets in R r , see Williams (1991), and the parameter space H 2 B r . According to Fonseca, Mexia, and Zmy slony (2007)…”

Section: Inducing Pivot Variablesmentioning

confidence: 99%

Chisquared and related inducing pivot variables: an application to orthogonal mixed models

Ferreira

Nunes

Fonseca

et al. 2019

Communications in Statistics - Theory and Methods

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“…Later on they were used by Seely (1970aSeely ( , b, 1971, Seely and Zyskind (1971), that named them as quadratic spaces, and used them to perform linear statistical inference. This approach had many interesting developments, see e. g. Zmyślony and Drygas (1992), VanLeeuwen et al (1998VanLeeuwen et al ( , 1999, and Fonseca et al (2006Fonseca et al ( , 2007Fonseca et al ( , 2008. This paper follows this trend.…”

Section: Algebras and Modelsmentioning

confidence: 99%

Inference for types and structured families of commutative orthogonal block structures

Carvalho

Mexia

Santos

et al. 2014

Metrika

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Your article is protected by copyright and all rights are held exclusively by Springer-Verlag Berlin Heidelberg. This e-offprint is for personal use only and shall not be selfarchived in electronic repositories. If you wish to self-archive your article, please use the accepted manuscript version for posting on your own website. You may further deposit the accepted manuscript version in any repository, provided it is only made publicly available 12 months after official publication or later and provided acknowledgement is given to the original source of publication and a link is inserted to the published article on Springer's website. The link must be accompanied by the following text: "The final publication is available at link.springer.com".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

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Jordan algebras, generating pivot variables and orthogonal normal models

Cited by 16 publications

References 12 publications

Estimability of variance components when all model matrices commute

Estimability of variance components when all model matrices commute

Chisquared and related inducing pivot variables: an application to orthogonal mixed models

Inference for types and structured families of commutative orthogonal block structures

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