Ricardo Covas

Structured Families of Models with Commutative Orthogonal Block Structures

Orthogonal Block Structure models are a well know class of models. In this paper we will work with a special class of such models, the ones with Commutative Orthogonal Block Structure, COBS. In this models the orthogonal projection matrix on the space spanned by the mean vector commutes with variance‐covariance matrix.The algebraic structure of such models will be studied and we will carry out inference for structured families of COBS where the models in these families correspond to the treatment of a base design.

Binary operations on Jordan algebras: An application to statistical inference in linear models

Necessary and sufficient conditions for the existence of best unbiased estimators (UMVUE) in Normal Linear Mixed Models, Y∼N(Xβ, ∑ i = 1kσi2Vi), are given. These conditions rely on the existence of Jordan algebras, in the sequence of [19], [6], [9], [7], [3] and [8]. We also take [2] in mind where lattices of Jordan algebras are presented for estimation purposes in these models and [1] where such estimation is exemplified.

A Fundamental Partition in Models with Commutative Orthogonal Block Structure

Models with commutative orthogonal block structure, COBS, constitute an interesting class of models with orthogonal block structure, OBS, in which the orthogonal projection matrix on the space Ω spanned by the mean vectors commute with the known pairwise orthogonal projection matrices Q1,…, Qm that figure in the expression of the variance‐covariance matrix V =  ∑ j = 1mγjQj of the model. We discuss the importance of the orthogonal partition Y = YΩ+YΩ⊥ where Y, YΩ and YΩ⊥ are the observation vectors and its orthogonal projection on Ω. and Ω⊥, the orthogonal complement on parameters estimation.

Structured families of quadratics

For the purpose of this work, we will consider that we have a base model for each of whose treatments we have a quadratic form. We study the action of the factors of the base model on the sub-vec of quadratic forms. We present F-tests for the hypothesis derived from these factors and study their relevance, through the definition of profiles. The estimators that are obtained are least squares estimators, hence the advantage of this approach.

Models with orthogonal block structure, with diagonal blockwise variance-covariance matrices

We intend to show that in the family of models with orthogonal block structure, OBS, we may single out those with blockwise diagonal variance-covariance matrices, DOBS. Namely we show that for every model with observation vector y with OBS, there is a model y°=Py, with P orthogonal which is DOBS and that the estimation of relevant parameters may be carried out for y°.

Construction of linear models: A framework based on commutative Jordan algebras

We show how to obtain the necessary structures for statistical analysis of the folllowing orthogonal models Y∼(1μ+∑iXiβi,∑jσj2Mj+σ2I). These structures rely on the existence of Jordan algebras, in the sequence of [24], [8], [12], [9], [5] and [10].

Mixed additive models

We consider mixed models y=∑i=0wXiβi with V(y)=∑i=1wθiMi Where Mi=XiXi⊤, i = 1, . . ., w, and µ = X0β0. For these we will estimate the variance components θ1, . . ., θw, aswell estimable vectors through the decomposition of the initial model into sub-models y(h), h ∈ Γ, with V(y(h))=γ(h)Ig(h)h∈Γ. Moreover we will consider L extensions of these models, i.e., y˚=Ly+e, where L=D (1n1, . . ., 1nw) and e, independent of y, has null mean vector and variance covariance matrix θw+1Iw, where w=∑i=1nwi.

ℬ-matrices and its applications to linear models

A new class of matrices is introduced, with interesting properties. Their use can be of most importance and the discussion of a possible use in statistical inference is presented. Namely the use in a class of models, the ones with mean driven balance, which are also defined.We point out the relevance of such models, when dealing with mixed models and the approach, for inference proposes, is the use of commutative Jordan algebras of symmetric matrices. The properties of symmetric B-matrices take a decisive roll in this approach.

Models with homoscedastic orthogonal partition, BQUE and mixed models

We consider models with homocedastic orthogonal partitions, such that each variance components is segregated into an orthogonal partition of Rm. By aproaching the estimators for the variance components and the analysis of mixed models, we intend to complete the study of uniformly best linear unbiased estimators (UBLUE) within these models.

Commutative orthogonal block structure: Orthogonal features

An interesting class of models with orthogonal block structure, OBS, are the ones with commutative orthogonal block structure, COBS. In such class of models the least square estimators are best linear unbiased estimators. In our approach we will consider the orthogonal partition Y = YΩ+YΩ⊥ where YΩ and YΩ⊥ are the orthogonal projections of the observations vector on the space Ω spanned by the mean vector and its orthogonal complement. When normality is assumed, we will obtain UMVUE estimators for the variance components in the family of estimators that depend only on YΩ⊥. When μ = X0β0 and the components of β0 correspond to the treatments of a fixed effects design Π, we show how to test for absence of effects and interaction for the factors in Π.