Binary operations on Jordan algebras and orthogonal normal models

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

doi:10.1016/j.laa.2006.03.045

Cited by 44 publications

(30 citation statements)

References 7 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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“…[19] showed that any CJA, A, has an unique basis, the pb(A) of A, constituted by POOPM. As stated by [5], Jordan algebras are used to present normal orthogonal models in a canonical form. Moreover:…”

Section: Commutative Jordan Algebrasmentioning

confidence: 99%

Estimation and Orthogonal Block Structure

Ferreirax¹,

Nunes²,

Ferreira³

et al. 2015

HJMS

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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].

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“…Seely called them quadratic spaces, [8], but for simplicity's sake we will retain the previous designation. This first use of quadratic subspaces in linear statistical inference led to the dissemination of several works where its use plays a central part, see e.g., [2,4,15] and [1].…”

Section: Commutative Jordan Algebrasmentioning

confidence: 99%

Complete and sufficient statistics and perfect families in orthogonal and error orthogonal normal models

2014

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We will discuss orthogonal models and error orthogonal models and their algebraic structure, using as background, commutative Jordan algebras. The role of perfect families of symmetric matrices will be emphasized, since they will play an important part in the construction of the estimators for the relevant parameters.Perfect families of symmetric matrices form a basis for the commutative Jordan algebra they generate. When normality is assumed, these perfect families of symmetric matrices will ensure that the models have complete and sufficient statistics. This will lead to uniformly minimum variance unbiased estimators for the relevant parameters.

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Binary operations on Jordan algebras and orthogonal normal models

Cited by 44 publications

References 7 publications

Estimability of variance components when all model matrices commute

Estimability of variance components when all model matrices commute

Estimation and Orthogonal Block Structure

Complete and sufficient statistics and perfect families in orthogonal and error orthogonal normal models

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