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- Chisquared and related inducing pivot variables: an application to orthogonal mixed modelsPublication . Ferreira, Dário; Ferreira, Sandra S.; Nunes, Célia; Fonseca, Miguel; Mexia, João T.We use chi-squared and related pivot variables to induce probability measures for model parameters, obtaining some results that will be useful on the induced densities. As illustration we considered mixed models with balanced cross nesting and used the algebraic structure to derive confidence intervals for the variance components. A numerical application is presented.
- Estimation of variance components in normal linear mixed models with additivityPublication . Ferreira, Dário; Ferreira, Sandra S.; Nunes, Célia; Mexia, João T.In this paper we use commutative Jordan Algebras to estimate variance components in linear mixed models. We apply the theory to a model in which three factors cross and one of the factors is additive to the other two.
- Confidence intervals for variance components in gauge capability studiesPublication . Ferreira, Dário; Ferreira, Sandra S.; Nunes, Célia; Oliveira, Teresa A.; Mexia, João T.We present a method, that uses pivot variables, which are functions of statistics and parameters, of constructing confidence intervals for variance components in gauge capability studies. As illustration we will consider a study on repeatability and reproducibility measures. Besides this the paper includes a simulation study demonstrating that in approximately 9500 out of 10000 simulations the 95% confidence interval covers the true value of the parameter.
- Maximum Likelihood Estimation Methods for Variance Components in Linear Non-Orthogonal Small Size Design ModelsPublication . Ferreira, Dário; Ferreira, Sandra S.; Nunes, Célia; Mexia, João T.We compare four Maximum Likelihood Estimation methods for estimating variance components in normal linear mixed models, in the case of unbalanced small size design models: The Newton-Raphson, the Triple Minimization, the Gradient and a method where the starting points for the Newton-Raphson are the estimates obtained with the Triple Minimization method.
- Estimation and Orthogonal Block StructurePublication . Ferreira, Sandra S.; Nunes, Célia; Ferreira, Dário; Moreira, Elsa; Mexia, João T.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].
- Estimation and incommutativity in mixed modelsPublication . Ferreira, Dário; Ferreira, Sandra S.; Nunes, Célia; Fonseca, Miguel; Silva, Adilson; Mexia, João T.In this paper we present a treatment for the estimation of variance components and estimable vectors in linear mixed models in which the relation matrices may not commute. To overcome this difficulty, we partition the mixed model in sub-models using orthogonal matrices. In addition, we obtain confidence regions and derive tests of hypothesis for the variance components. A numerical example is included. There we illustrate the estimation of the variance components using our treatment and compare the obtained estimates with the ones obtained by the ANOVA method. Besides this, we also present the restricted and unrestricted maximum likelihood estimates.
- Segregation and intrinsec restrictions on canonic variance componentsPublication . Ferreira, Dário; Ferreira, Sandra S.; Nunes, Célia; Mexia, João T.This paper deals with the estimability of variance components, in mixed models, when the dimension of the commutative algebra, spanned by all possible variance-covariance matrices, is greater than the number of linearly independente unknown variance components. As example we present an application to a random three-factor crossed-model.
- Estimation in mixed models through three step minimizationPublication . Ferreira, Dário; Ferreira, Sandra S.; Nunes, Célia; Mexia, João T.The aim of this article is to present an estimation procedure for both fixed effects and variance components in linear mixed models. This procedure consists of a maximum likelihood method which we call Three Step Minimization, TSM. The major contribution of this method is that when variances tend to be null standard algorithms behave badly, unlike the TSM method, which uses a grid search algorithm in a compact set. A numerical application with real and simulated data is provided.
- Tests and relevancies for the hypotheses of an orthogonal family in a model with orthogonal block structurePublication . Ferreira, Dário; Ferreira, Sandra S.; Nunes, Célia; Mexia, João T.A model has an orthogonal block structure if it has, as covariance matrix, a linear combination of pairwise orthogonal projection matrices, that add up to the iden- tity matrix. The range space of these matrices are associated to hypotheses of an orthogonal family. In this paper we show how to obtain tests for these hypotheses when normality is assumed and how to consider their relevance when normality is discarded. Besides the notion of relevance, we formulate hypotheses in a general way that may be applied to models with orthogonal block structure, whose factors may have xed and/or random e ects. The results are applied to prime basis factorial models and an example is presented.