This thesis develops methodologies for the construction of various types of optimal designs with applications in maximum likelihood estimation and factorial structure design. The methodologies are applied to some real data sets throughout the thesis. We start with a broad review of optimal design theory including various types of optimal designs along with some fundamental concepts. We then consider a class of optimization problems and determine the optimality conditions. An important tool is the directional derivative of a criterion function. We study extensively the properties of the directional derivatives. In order to determine the optimal designs, we consider a class of multiplicative algorithms indexed by a function, which satisfies certain conditions. The most important and popular design criterion in applications is D-optimality. We construct such designs for various regression models and develop some useful strategies for better convergence of the algorithms. The remaining thesis is devoted to some important applications of optimal design theory. We first consider the problem of determining maximum likelihood estimates of the cell probabilities under the hypothesis of marginal homogeneity in a square contingency table. We formulate the Lagrangian function and remove the Lagrange parameters by substitution. We then transform the problem to one of maximizing some functions of the cell probabilities simultaneously. We apply this problem to some real data sets, namely, a US Migration data, and a data on grading of unaided distance vision. We solve another estimation problem to determine the maximum likelihood estimation of the parameters of the latent variable models such as Bradley-Terry model where the data come from a paired comparisons experiment. We approach this problem by considering the observed frequency having a binomial distribution and then replacing the binomial parameters in terms of optimal design weights. We apply this problem to a data set from American League Baseball Teams. Finally, we construct some optimal structure designs for comparing test treatments with a control. We introduce different structure designs and establish their properties using the incidence and characteristic matrices. We also develop methods of obtaining optimal R-type structure designs and show how such designs are trace, A- and MV-optimal. / October 2016

<p>Previous papers have elaborated formal gradient analysis for spatial processes, focusing on the distribution theory for directional derivatives associated with a response variable assumed to follow a Gaussian process model. In the current work, these ideas are extended to additionally accommodate one or more continuous covariate(s) whose directional derivatives are of interest and to relate the behavior of the directional derivatives of the response surface to those of the covariate surface(s). It is of interest to assess whether, in some sense, the gradients of the response follow those of the explanatory variable(s), thereby gaining insight into the local relationships between the variables. The joint Gaussian structure of the spatial random effects and associated directional derivatives allows for explicit distribution theory and, hence, kriging across the spatial region using multivariate normal theory. The gradient analysis is illustrated for bivariate and multivariate spatial models, non-Gaussian responses such as presence-absence and point patterns, and outlined for several additional spatial modeling frameworks that commonly arise in the literature. Working within a hierarchical modeling framework, posterior samples enable all gradient analyses to occur as post model fitting procedures.</p> / Dissertation

Generalized linear mixture models (GLMM) are widely used in statistical applications to model count and binary data. We consider the problem of nonparametric likelihood estimation of mixing distributions in GLMM's with multiple random effects. The log-likelihood to be maximized has the general form l(G)=Σi log∫f(yi,γ) dG(γ) where f(.,γ) is a parametric family of component densities, yi is the ith observed response dependent variable, and G is a mixing distribution function of the random effects vector γ defined on Ω. The literature presents many algorithms for maximum likelihood estimation (MLE) of G in the univariate random effect case such as the EM algorithm (Laird, 1978), the intra-simplex direction method, ISDM (Lesperance and Kalbfleish, 1992), and vertex exchange method, VEM (Bohning, 1985). In this dissertation, the constrained Newton method (CNM) in Wang (2007), which fits GLMM's with random intercepts only, is extended to fit clustered datasets with multiple random effects. Owing to the general equivalence theorem from the geometry of mixture likelihoods (see Lindsay, 1995), many NPMLE algorithms including CNM and ISDM maximize the directional derivative of the log-likelihood to add potential support points to the mixing distribution G. Our method, Direct Search Directional Derivative (DSDD), uses a directional search method to find local maxima of the multi-dimensional directional derivative function. The DSDD's performance is investigated in GLMM where f is a Bernoulli or Poisson distribution function. The algorithm is also extended to cover GLMM's with zero-inflated data. Goodness-of-fit (GOF) and selection methods for mixed models have been developed in the literature, however their application in models with nonparametric random effects distributions is vague and ad-hoc. Some popular measures such as the Deviance Information Criteria (DIC), conditional Akaike Information Criteria (cAIC) and R2 statistics are potentially useful in this context. Additionally, some cross-validation goodness-of-fit methods popular in Bayesian applications, such as the conditional predictive ordinate (CPO) and numerical posterior predictive checks, can be applied with some minor modifications to suit the non-Bayesian approach. / Graduate / 0463 / rabihsaab@gmail.com

Generalized linear mixture models (GLMM) are widely used in statistical applications to model count and binary data. We consider the problem of nonparametric likelihood estimation of mixing distributions in GLMM's with multiple random effects. The log-likelihood to be maximized has the general form l(G)=Σi log∫f(yi,γ) dG(γ) where f(.,γ) is a parametric family of component densities, yi is the ith observed response dependent variable, and G is a mixing distribution function of the random effects vector γ defined on Ω. The literature presents many algorithms for maximum likelihood estimation (MLE) of G in the univariate random effect case such as the EM algorithm (Laird, 1978), the intra-simplex direction method, ISDM (Lesperance and Kalbfleish, 1992), and vertex exchange method, VEM (Bohning, 1985). In this dissertation, the constrained Newton method (CNM) in Wang (2007), which fits GLMM's with random intercepts only, is extended to fit clustered datasets with multiple random effects. Owing to the general equivalence theorem from the geometry of mixture likelihoods (see Lindsay, 1995), many NPMLE algorithms including CNM and ISDM maximize the directional derivative of the log-likelihood to add potential support points to the mixing distribution G. Our method, Direct Search Directional Derivative (DSDD), uses a directional search method to find local maxima of the multi-dimensional directional derivative function. The DSDD's performance is investigated in GLMM where f is a Bernoulli or Poisson distribution function. The algorithm is also extended to cover GLMM's with zero-inflated data. Goodness-of-fit (GOF) and selection methods for mixed models have been developed in the literature, however their application in models with nonparametric random effects distributions is vague and ad-hoc. Some popular measures such as the Deviance Information Criteria (DIC), conditional Akaike Information Criteria (cAIC) and R2 statistics are potentially useful in this context. Additionally, some cross-validation goodness-of-fit methods popular in Bayesian applications, such as the conditional predictive ordinate (CPO) and numerical posterior predictive checks, can be applied with some minor modifications to suit the non-Bayesian approach. / Graduate / 0463 / rabihsaab@gmail.com

Submitted by Johnny Rodrigues (johnnyrodrigues@ufcg.edu.br) on 2018-08-08T20:21:22Z No. of bitstreams: 1 ARTHUR GILZEPH FARIAS ALMEIDA - DISSERTAÇÃO PPGMAT 2013..pdf: 508810 bytes, checksum: 02ca89b269a1cb82e4ba0a5d102acff9 (MD5) / Made available in DSpace on 2018-08-08T20:21:22Z (GMT). No. of bitstreams: 1 ARTHUR GILZEPH FARIAS ALMEIDA - DISSERTAÇÃO PPGMAT 2013..pdf: 508810 bytes, checksum: 02ca89b269a1cb82e4ba0a5d102acff9 (MD5) Previous issue date: 2013-10 / CNPq / Neste trabalho estudamos a existência de, pelo menos, três soluções distintas para dois problemas de inclusão diferencial. Para isto, faremos uso da teoria da análise convexa para funcionais localmente Lipschitz, bem como métodos variacionais. / In this work we study the existence of, at least, three distinct solutions to two problems of differential inclusion. For this, we use the theory of convex functional analysis Lipschitz locally, and variational methods.

This thesis deals with the introduction of function of two variables and differential calculus of this function. This work should serve as a textbook for students of elementary school's teacher. Each chapter contains a summary of basic concepts and explanations of relationships, then solved model exercises of the topic and finally the exercises, which should solve the student himself. Thesis have transmit to students basic knowledges of differential calculus of functions of two variables, including practical knowledges.