A Characterization of Best Linear Unbiased Estimators in the General Linear Model

Zmyślony, Roman

doi:10.1007/978-1-4615-7397-5_27

Cited by 32 publications

(26 citation statements)

References 3 publications

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“…Now, in COBS, matrices T and V commute whatever the variance components, which, see Theorem 2 in [19], is a necessary and sufficient condition for LSE to be UBLUE. A similar result can also be found in [20].…”

Section: Elasupporting

confidence: 86%

Commutative orthogonal block structure and error orthogonal models

Carvalho¹,

Mexia²,

Santos³

2012

ELA

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Abstract.A model has orthogonal block structure, OBS, if it has variance-covariance matrix that is a linear combination of known pairwise orthogonal orthogonal projection matrices that sum to the identity matrix. These models were introduced by Nelder is 1965, and continue to play an important part in randomized block designs.Two important types of OBS are related, and necessary and sufficient conditions for model of one type belonging to the other are determined.The first type, is that of models with commutative orthogonal block structure in which T, the orthogonal projection matrix on the space spanned by the mean vector, commutes with the orthogonal projection matrices in the expression of the variance-covariance matrix.The second type, is that of error orthogonal models.These results open the possibility of deepening the study of the important class of models with OBS.

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Section: Elasupporting

confidence: 86%

Commutative orthogonal block structure and error orthogonal models

Carvalho¹,

Mexia²,

Santos³

2012

ELA

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“…This result is interesting since in COBS the LSE are UBLUE, being BLUE whatever θ, see [23]. Thus we validate the above proposition showing that our "sub-optimal estimator" is "optimal" when the model enjoys COBS.…”

Section: Proposition When the Model Has Cobs Theψ Are Lsesupporting

confidence: 79%

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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“…The importance of Assumption 10 is shown by the following theorem, which can be found in [20,22,24,26,33,44], for example. More recently, Assumption 10 has been called 'equivalent estimation'; see, for example, [25,30,43].…”

Section: Assumption 9 [Commutativity Of Pattern Matrices] Assumptionmentioning

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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A Characterization of Best Linear Unbiased Estimators in the General Linear Model

Cited by 32 publications

References 3 publications

Commutative orthogonal block structure and error orthogonal models

Commutative orthogonal block structure and error orthogonal models

Estimation and Orthogonal Block Structure

Estimability of variance components when all model matrices commute

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