Francisco Carvalho

Models with Stair Nesting

When we work with balanced nested designs, the usual approach forces us to divide repeatedly the plots, so we have few degrees of freedom for the first levels. As an alternative we have the stair nested designs. Stair nesting leads to very light models since we can work with fewer observations and the amount of information for the different factors is more evenly distributed. This is in fact a big advantage when comparing with balanced nesting, since stair nesting will allow experiments that will become cheaper, due to the less number of observations involved, or with the same resources produce more experiments. For models with stair nesting it is easy to carry out inference, as we shall see. It is interesting to point out that stair nested designs although being orthogonal, with the advantages associated to that condition, are not balanced.

Estimation in Models with Commutative Orthogonal Block Structure

A model with variance-covariance matrix V = Σi=1vσi2Pi, where P1, …, Pv are known pairwise orthogonal orthogonal projection matrices, will have Orthogonal Block Structure with variance components σ12, …, σv2. Moreover, if matrices P1, …, Pv commute with the orthogonal projection matrix T on the space spanned by the mean vector, the model will have Commutative Orthogonal Block Structure (COBS). In this paper we will use Commutative Jordan Algebras to study the algebraic properties of these models as well as optimal estimators. We show that once normality is assumed, sufficient complete statistics are obtained and estimators are Uniformly Minimum Variance Unbiased Estimators.

COMMUTATIVE ORTHOGONAL BLOCK STRUCTURE AND ERROR ORTHOGONAL MODELS

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.

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.

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].

Estimation for large non-centrality parameters

We introduce the concept of estimability for models for which accurate estimators can be obtained for the respective parameters. The study was conducted for model with almost scalar matrix using the study of estimability after validation of these models. In the validation of these models we use F statistics with non centrality parameter τ=‖λ‖2σ2 when this parameter is sufficiently large we obtain good estimators for λ and α so there is estimability. Thus, we are interested in obtaining a lower bound for the non-centrality parameter. In this context we use for the statistical inference inducing pivot variables, see Ferreira et al. 2013, and asymptotic linearity, introduced by Mexia & Oliveira 2011, to derive confidence intervals for large non-centrality parameters (see Inacio et al. 2015). These results enable us to measure relevance of effects and interactions in multifactors models when we get highly statistically significant the values of F tests statistics.

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.