Inference for types and structured families of commutative orthogonal block structures

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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“…. , M w ) with Assumption 10 is called 'commutative orthogonal block structure' (COBS) in [10,15,19,29].…”

Section: Examplementioning

confidence: 99%

Estimability of variance components when all model matrices commute

Bailey

Ferreira

Nunes

2016

Linear Algebra and its Applications

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“…where 𝜷 is fixed and the u 1 , …, u w are independent, with null mean vectors and variance-covariance matrices 18 if and only if they are diagonalized by the same orthogonal matrix P. Then, M ⊂ V(P), with V(P) the family of matrices diagonalized by P. This family is a Commutative Jordan algebra, CJA, being a linear space constituted by symmetric matrices that commute and containing the squares of its matrices, see, e.g., Carvalho et al 19 Since the intersection of CJA gives CJA and M ⊂ V(P), the intersection of CJA containing M gives the least CJA that contains M. This will be the CJA (M). According to Seely, 20 any CJA has a unique basis constituted by POOPM, the principal basis, pb().…”

Section: Variance Componentsmentioning

confidence: 99%

Confidence intervals and ellipsoids for estimable functions and estimable vectors in models with orthogonal block structure

Ferreira

Nunes

Mexia

2022

Math Methods in App Sciences

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Sub‐models of mixed linear models are considered. The independence of these sub‐models leads to sufficient statistics for the parameters relevant for their densities. Using pivot variables, confidence regions are obtained as well hypothesis testing for variance components, estimable functions, and estimable vectors. In addition, to compare the estimators and the models, we present the histograms with the empirical joint densities for positive and negative parts of the estimators. The figures, for the two‐dimensional charts, contain the corresponding UMVUE and are all unimodal with the UMVUE near the mode. The nearness of the estimators and the modes validates the presented methodology and allows the safe use of induced densities. A numerical example applied to real data is presented.

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“…The inference for this family of regressions is centered on the vectors of coefficients or, more generally, on estimable vectors, leading to interesting results, see. [15][16][17][18][19][20][21] These models, in a structured family, correspond to the treatments of a base design with fixed effects. The most interesting case is when the absence of effects and interactions for the factors in the base design are associated to the spaces of an orthogonal partition…”

Section: Structured Familiesmentioning

confidence: 99%

Isolated and structured families of models for stochastic symmetric matrices

Dias

Santos

Mexia

2021

Comp and Math Methods

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Stochastic symmetric matrices with a dominant eigenvalue, 𝛂,can be written as the sum of λ𝛂𝛂 t (where λ is the first eigenvalue), with a symmetric error matrix E. The information in the stochastic matrix will be condensed in its structured vectors, λ𝛂, and the sum of square of residues, V. When the matrices of a family correspond to the treatments of a base design, we say the family is structured. The action of the factors, which are considered in the base design, on the structure vectors of the family matrices will be analyzed. We use ANOVA (Analysis of Variance) and related techniques, to study the action under linear combinations of the components of structure vectors of the m matrices of the model. Orthogonal models with m treatments are associated to orthogonal partitions. The hypothesis to be tested, on the action of the factors in the base design, will be associated to the spaces in the orthogonal partitions. We will show how to carry out transversal and longitudinal analysis for families of stochastic symmetric matrices with dominant eigenvalue associated to orthogonal models.

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Inference for types and structured families of commutative orthogonal block structures

Cited by 12 publications

References 21 publications

Estimability of variance components when all model matrices commute

Estimability of variance components when all model matrices commute

Confidence intervals and ellipsoids for estimable functions and estimable vectors in models with orthogonal block structure

Isolated and structured families of models for stochastic symmetric matrices

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