O objetivo deste trabalho é fazer uma revisão dos modelos não lineares de família exponencial (Cordeiro & Paula (1989); Wei (1998)) para respostas independentes e apresentar possíveis extensões para o caso de dados correlacionados. Inicialmente são apresentados exemplos ilustrativos, alguns dos quais são reanalizados ao longo do texto. Em seguida são discutidos procedimentos de estimação e testes de hipóteses, tais como apresentação de um processo de estimação que pode ser adaptado ao processo iterativo usado na classe dos modelos lineares generalizados, e alguns resultados assintóticos. Técnicas usuais de diagnóstico, como pontos de alavanca, análise de resíduos e diagnóstico de influência são adaptados para a classe dos modelos não lineares de família exponencial. Extensões para a classe dos modelos não lineares com resposta binomial negativa são também apresentadas. Finalmente, são consideradas duas possíveis extensões dos modelos não lineares de família exponencial para dados correlacionados, através de equações de estimação generalizadas e através de modelagem mista em que efeitos aleatórios em forma linear são adicionados ao componente não linear da parte sistemática do modelo conforme sugerido recentemente por Tang et al. (2006a). / The aim of this work is to present a review of the exponential family nonlinear models (Cordeiro & Paula (1989); Wei (1998)) for independent responses and to present possible extensions for the case of correlated data. Firstly, ilustrative examples are presented with some of them being reanalyzed along the text. Then, estimation and hypothesis testing procedures, such as the presentation of an iterative process adapted from the one of generalized linear models, and some asymptotic results are discussed. Useful diagnostic techniques, as calculation of leverage measures, residual analysis and influence diagnostics are adapted for the class of exponential family nonlinear models. Extensions to nonlinear negative binomial models are also presented. Finally, two possible extensions for correlated data are considered, by using generalized estimating equations and mixed modeling in which linear random effects are added into the systematic component together with the nonlinear function, as suggested by Tang et al. (2006a).

O objetivo deste trabalho é fazer uma revisão dos modelos não lineares de família exponencial (Cordeiro & Paula (1989); Wei (1998)) para respostas independentes e apresentar possíveis extensões para o caso de dados correlacionados. Inicialmente são apresentados exemplos ilustrativos, alguns dos quais são reanalizados ao longo do texto. Em seguida são discutidos procedimentos de estimação e testes de hipóteses, tais como apresentação de um processo de estimação que pode ser adaptado ao processo iterativo usado na classe dos modelos lineares generalizados, e alguns resultados assintóticos. Técnicas usuais de diagnóstico, como pontos de alavanca, análise de resíduos e diagnóstico de influência são adaptados para a classe dos modelos não lineares de família exponencial. Extensões para a classe dos modelos não lineares com resposta binomial negativa são também apresentadas. Finalmente, são consideradas duas possíveis extensões dos modelos não lineares de família exponencial para dados correlacionados, através de equações de estimação generalizadas e através de modelagem mista em que efeitos aleatórios em forma linear são adicionados ao componente não linear da parte sistemática do modelo conforme sugerido recentemente por Tang et al. (2006a). / The aim of this work is to present a review of the exponential family nonlinear models (Cordeiro & Paula (1989); Wei (1998)) for independent responses and to present possible extensions for the case of correlated data. Firstly, ilustrative examples are presented with some of them being reanalyzed along the text. Then, estimation and hypothesis testing procedures, such as the presentation of an iterative process adapted from the one of generalized linear models, and some asymptotic results are discussed. Useful diagnostic techniques, as calculation of leverage measures, residual analysis and influence diagnostics are adapted for the class of exponential family nonlinear models. Extensions to nonlinear negative binomial models are also presented. Finally, two possible extensions for correlated data are considered, by using generalized estimating equations and mixed modeling in which linear random effects are added into the systematic component together with the nonlinear function, as suggested by Tang et al. (2006a).

Doctor of Philosophy / Department of Statistics / James Neill / The problem of testing for lack of fit in exponential family regression models is considered. Such nonlinear models are the natural extension of Normal nonlinear regression models and generalized linear models. As is usually the case, inadequately specified models have an adverse impact on statistical inference and scientific discovery. Models of interest are curved exponential families determined by a sequence of predictor settings and mean regression function, considered as a sub-manifold of the full exponential family. Constructed general alternative models are based on clusterings in the mean parameter components and allow likelihood ratio testing for lack of fit associated with the mean, equivalently natural parameter, for a proposed null model. A maximin clustering methodology is defined in this context to determine suitable clusterings for assessing lack of fit. In addition, a geometrically motivated goodness of fit test statistic for exponential family regression based on the information metric is introduced. This statistic is applied to the cases of logistic regression and Poisson regression, and in both cases it can be seen to be equal to a form of the Pearson chi[superscript]2 statistic. This same statement is true for multinomial regression. In addition, the problem of testing for equal means in a heteroscedastic Normal model is discussed. In particular, a saturated 3 parameter exponential family model is developed which allows for equal means testing with unequal variances. A simulation study was carried out for the logistic and Poisson regression models to investigate comparative performance of the likelihood ratio test, the deviance test and the goodness of fit test based on the information metric. For logistic regression, the Hosmer-Lemeshow test was also included in the simulations. Notably, the likelihood ratio test had comparable power with that of the Hosmer-Lemeshow test under both m- and n-asymptotics, with superior power for constructed alternatives. A distance function defined between densities and based on the information metric is also given. For logistic models, as the natural parameters go to plus or minus infinity, the densities become more and more deterministic and limits of this distance function are shown to play an important role in the lack of fit analysis. A further simulation study investigated the power of a likelihood ratio test and a geometrically derived test based on the information metric for testing equal means in heteroscedastic Normal models.

This thesis identifies and extends techniques that can be linked to the principle of maximum entropy (maxent) and applied to parameter estimation in machine learning and statistics. Entropy functions based on deformed logarithms are used to construct Bregman divergences, and together these represent a generalization of relative entropy. The framework is analyzed using convex analysis to charac- terize generalized forms of exponential family distributions. Various connections to the existing machine learning literature are discussed and the techniques are applied to the problem of non-negative matrix factorization (NMF).

The subject of this thesis is the maximization of the information divergence from an exponential family on a finite set, a problem first formulated by Nihat Ay. A special case is the maximization of the mutual information or the multiinformation between different parts of a composite system. My thesis contributes mainly to the mathematical aspects of the optimization problem. A reformulation is found that relates the maximization of the information divergence with the maximization of an entropic quantity, defined on the normal space of the exponential family. This reformulation simplifies calculations in concrete cases and gives theoretical insight about the general problem. A second emphasis of the thesis is on examples that demonstrate how the theoretical results can be applied in particular cases. Third, my thesis contain first results on the characterization of exponential families with a small maximum value of the information divergence.

Submitted by (ana.araujo@ufrpe.br) on 2016-05-19T14:39:16Z No. of bitstreams: 1 Alessandro Henrique da Silva Santos.pdf: 536044 bytes, checksum: 011926984df2097b6901e78e8a50d28e (MD5) / Made available in DSpace on 2016-05-19T14:39:16Z (GMT). No. of bitstreams: 1 Alessandro Henrique da Silva Santos.pdf: 536044 bytes, checksum: 011926984df2097b6901e78e8a50d28e (MD5) Previous issue date: 2009-02-27 / The exponential family nonlinear models are an extension of the generalized models, opening various options for the distribution of the variable answer and allowing larger flexibility for the connection between the average and the systematic component. These models, for being less restrictive, having been used to model several phenomena in the nature. To estimate the parameters of these models, several procedures are proposed. Usually, the method of maximum likelihood, that has asymptotic properties of order n-1, where n is the size of the sample, it is the used. In this work we will make a general approach to the no-linear models of the exponential family. The theory of the exponential family will be introduced presenting the function of density of probability, function cumulantes geratriz, likelihood function, likelihood ratio and deviation of the model; such presented results will facilitate and/or they will be necessary in the understanding of what will be done for the nonlinear models of the exponential family. The exponential family nonlinear models will be defined by presenting the suppositions of the model, its likelihood function and the algorithm for the estimate of the parameters. We will make the approach of the diagnosis analysis and of influence of the exponential family nonlinear models. Finally, we will present some applications and we will show the efficiency and importance in the use of this class, once several phenomena present nonlinear behavior. / Os modelos não-lineares da família exponencial são uma extensão dos modelos generalizados, abrindo um leque de opções para a distribuição da variável resposta e permitindo maior flexibilidade para a ligação entre a média e a componente sistemática. Estes modelos, por serem menos restritivos, têm sido utilizados para modelar diversos fenômenos na natureza. Para estimar os parâmetros destes modelos, vários procedimentos são propostos. Usualmente, o método de máxima verossimilhança, que tem propriedades assintóticas de ordem n-1, onde n é o tamanho da amostra, é o mais utilizado. Neste trabalho faremos uma abordagem geral dos modelos não-lineares da família exponencial. Será introduzida a teoria da família exponencial sendo apresentada a função de densidade de probabilidade, função geratriz de cumulantes, função de verossimilhança, razão de verossimilhança e desvio do modelo; tais resultados apresentados facilitarão e/ou serão necessários na compreensão do que será feito para os modelos não-lineares da família exponencial. Será definido o modelo não-linear da família exponencial sendo apresentadas as suposições do modelo, sua função de verossimilhança e algoritmo da estimação dos parâmetros. Faremos a abordagem da análise de diagnóstico e de influência dos modelos não-lineares da família exponencial. Por fim, faremos aplicações e mostraremos a eficiência e importância na utilização desta classe, uma vez que diversos fenômenos apresentam comportamento não-linear.

In 2018, Raul Matsushita, Donald Pianto, Bernardo B. De Andrade, Andre Can§ado & Sergio Da Silva published a paper titled âTouchard distributionâ, which presented a model that is a two-parameter extension of the Poisson distribution. This model has its normalizing constant related to the Touchard polynomials, hence the name of this model. This diploma thesis is concerned with the properties of the Touchard distribution for which delta is known. Two asymptotic tests based on two different statistics were carried out for comparison in a Touchard model with two independent samples, supported by simulations in R.

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This thesis deals with the bootstrap method. Three decades after the seminal paper by Bradly Efron, still the horizons of this method need more exploration. The research presented herein has stepped into different fields of statistics where the bootstrap method can be utilized as a fundamental statistical tool in almost any application. The thesis considers various statistical problems, which is explained briefly below. Bootstrap method: A comparison of the parametric and the nonparametric bootstrap of variance is presented. The bootstrap of ranked set sampling is dealt with, as well as the wealth of theories and applications on the RSS bootstrap that exist nowadays. Moreover, the performance of RSS in resampling is explored. Furthermore, the application of the bootstrap method in the inference of contingency table test is studied. Coefficient of variation: This part shows the capacity of the bootstrap for inferring the coefficient of variation, a task which the asymptotic method does not perform very well. Information theory: There are few works on the study of information theory, especially on the inference of entropy. The papers included in this thesis try to achieve the inference of entropy using the bootstrap method.