Given a bivariate sample [special characters omitted], the rth order statistic [special characters omitted] is the rth smallest value of the X's and the rth concomitant [special characters omitted] is the Y value that accompanies [special characters omitted]. Order statistics are widely used, and their stochastic order and dependence properties have been studied extensively. In this dissertation, we show that if X and Y are positively dependent, then the concomitants satisfy certain stochastic order relations and positive dependence properties as well, at least in the case where the vectors ( Xi, Yi) are independent and identically distributed and come from an absolutely continuous distribution. It, is shown that if Y is stochastically increasing in X, the concomitants increase in multivariate stochastic order, and the entire vector of concomitants [special characters omitted] is multivariate associated. If the conditional hazard rate function of Y given X, [special characters omitted] is decreasing in x, [special characters omitted] is multivariate right corner set increasing. If X and Y are totally positive dependent of order 2, then [special characters omitted] is multivariate totally positive dependent of order 2, and the univariate concomitants [special characters omitted] increase in likelihood ratio order as r increases. Concomitants have not previously been studied in the discrete case much because, unlike the continuous case, the probability that two order statistics [special characters omitted] and [special characters omitted] are equal is positive; thus, the concomitants are not immediately determinable. Here we introduce a way of assigning concomitants in the discrete case and prove two related results for that case.

The q-state Curie-Weiss-Potts model, where q $\ge$ 3 is an integer, is a useful statistical mechanical model. It is an exponential family parametrized by the inverse absolute temperature $\beta$ and the external magnetic field h. As this dissertation shows, the model has a fascinating probabilistic structure. For q = 2, the model is equivalent to the classical Curie-Weiss model. The first part of the dissertation studies limit theorems for the empirical vector, $L\sb{n}(\omega)$, of the model. These limits include the law of large numbers, a central limit theorem when $\beta$ $<$ $\beta\sb{c}$ and h = 0, and a conditional central limit theorem when $\beta \ge \beta\sb{c}$ and h = 0, where $\beta\sb{c}$ is the critical inverse temperature. Also a central limit theorem with random centering is proved. The phase transition at $\beta\sb{c}$ is first-order, in contrast to a second-order phase transition in the classical Curie-Weiss model. All these limit theorems imply similar limits for the sample mean $n\sp{-1} S\sb{n}(\omega)$. Some limit theorems for the classical Curie-Weiss model are also presented. The second part of the dissertation studies the large sample behavior of the maximum likelihood estimator, $\ h\sb{n}$, of the external magnetic field h. I will study $\ h\sb{n}$ when $\beta$ is given and the true value of h is known to be 0. Under suitable conditioning, it is found that $\ h\sb{n}$ exists and is unique. It is also found that under suitable conditioning, $\sqrt{n}{\ h\sb{n}}$ has a normal limit when $\beta<\beta\sb{c}$ and a discontinuous limit when $\beta\ge\beta\sb{c}$. Despite this discontinuous limit, $\ h\sb{n}$ is always consistent for h whenever $\beta$ is given. I will also summarize some results on the maximum likelihood estimator, $\\beta\sb{n}$, of the inverse absolute temperature $\beta$. These results have been proved in Ellis-Wang (1990b).

Thesis (Ed.D.)--Teachers College, Columbia University, 1953. / Typescript. Type C project. Includes tables. Sponsor: H. F. Fehr. Dissertation Committee: M. C. Rosskopf, H. M. Walker. Includes bibliographical references (leaves 126-132, 162-167).

Thesis (Ph. D.)--University of Wisconsin--Madison, 1969. / Typescript. Vita. eContent provider-neutral record in process. Description based on print version record. Includes bibliographical references.

Generalized Estimating Equations (GEE) and Mixed Effects Logistic Models have become popular methods for analyzing clustered binary data when regression is the primary focus. However, methods to assess the goodness of fit of the fitted models are not well developed. Recent work concerning the independence case of ordinary logistic regression provides the basis for a newly computed mean and variance for the Pearson chi-square statistic and the unweighted sums of squares statistic for the clustered binary data case. A simulation study was conducted to evaluate the performance of the Pearson chi-square statistic. the unweighted sums of squares statistic. as well as the Hosmer-Lemeshow statistic for the GEE model. Another simulation study was conducted to evaluate the performance of the several versions of the Pearson chi-square statistic. an unweighted sums of squares statistic, a Deviance statistic, as well as the Hosmer-Lemeshow statistic far the mixed effects logistic model. The factors that were varied were the number of clusters, the number of observations within a cluster, the magnitude of the correlation or random effect, and the number and type of covariates included in the model. The proposed methods were then applied to real data sets. Overall, several of the statistics that were examined had a satisfactory performance of a Type I error rate and are potentially effective in evaluating goodness of fit under certain conditions. Limitations are discussed.

A large collection of digital images of natural scenes provides a database for analyzing and modeling small scene patches (e.g., 2 x 2) referred to as natural microimages. A pivotal finding is the stability of the empirical microimage distribution across scene samples and with respect to scaling. With a view toward potential applications (e.g. classification, clutter modeling, segmentation), we present a hierarchy of microimage probability models which capture essential local image statistics. Tools from information theory, algebraic geometry and of course statistical hypothesis testing are employed to assess the “match” between candidate models and the empirical distribution. Geometric symmetries play a key role in the model selection process. One central result is that the microimage distribution exhibits reflection and rotation symmetry and is well-represented by a Gibbs law with only pairwise interactions. However, the acceptance of the up-down reflection symmetry hypothesis is borderline and intensity inversion symmetry is rejected. Finally, possible extensions to larger patches via entropy maximization and to patch classification via vector quantization are briefly discussed.

This dissertation addresses three basic issues that arise in the use of Gaussian Markov random fields (GMRFs) in image modelling: the multi-resolution properties, the valid parameter space, and the existence of Maximum Likelihood (ML) and Maximum Entropy (ME) parameter estimates. For the multi-resolution properties, we study GMRFs under two types of resolution transformations, Sampling and Block-to-Point. We show that under both these transformations the coarser level fields are non-Markov, and obtain exact descriptions for their covariances and power spectra. To approximate the coarser level non-Markov fields as GMRFs, we propose a new methodology called the Covariance Invariance Approximation (CIA) and study its measure-theoretic properties. We argue that CIA is better suited to image processing than the free-energy based approximations used in renormalization group studies. On the valid parameter space issue, for both 1-D infinite-length GM processes and 2-D infinite-lattice GMRFs, we present a complete procedure for verifying the validity of a given set of parameters. We illustrate this result by applying it to second-order fields in both 1-D and 2-D, and obtain an explicit and simple description of the respective parameter spaces. We observe that in both these examples, the valid parameter space is considerably larger than the space implied by the previously known sufficient condition. For both 1-D and 2-D finite-lattice fields, we show that the valid parameter space does not admit a simple description. The infinite-lattice conditions, however, provide a tight lower-bound approximation to the valid parameter space of finite-lattice fields. Finally, we consider the existence of the ML and the ME estimates for GMRF parameters. The existence of ME estimates is closely related to the extendibility of covariance sequences. Using this fact in conjunction with our results on the valid parameter space of GMRFs, we obtain analytical and computational solutions to the existence problem. For several examples, we obtain an explicit set of conditions that ascertain extendibility and hence existence. For the general case, we propose a cutting-plane algorithm as an alternative to the two numerical procedures that already exist for determining extendibility, namely, the linear programming algorithm and expanding-hull algorithm. Next, we explore the duality between the valid parameter space of GMRFs and the space of extendible covariances, and their relationships with the space of admissible covariances for finite-size data sequences. Using duality, we also relate the existence of ML estimates to extendibility and show that the existence of ML estimates would have to be ascertained through a computationally intensive linear programming procedure. Finally, we present some results regarding the extendibility of covariances over increasing window sizes.

The purpose of this study is to investigate and understand data which are grouped into categories. Various statistical methods was studied for categorical binary responses to investigate the causes of death from diarrhoea in South Africa. Data collected included death type, sex, marital status, province of birth, province of death, place of death, province of residence, education status, smoking status and pregnancy status. The objective of this thesis is to investigate which of the above explanatory variables was most affected by diarrhoea in South Africa. To achieve this objective, different sample survey data analysis techniques are investigated. This includes sketching bar graphs and using several statistical methods namely, logistic regression, surveylogistic, generalised linear model, generalised linear mixed model, and generalised additive model. In the selection of the fixed effects, a bar graph is applied to the response variable individual profile graphs. A logistic regression model is used to identify which of the explanatory variables are more affected by diarrhoea. Statistical applications are conducted in SAS (Statistical Analysis Software). Hosmer and Lemeshow (2000) propose a statistic that they show, through simulation, is distributed as chi‐square when there is no replication in any of the subpopulations. Due to the similarity of the Hosmer and Lemeshow test for logistic regression, Parzen and Lipsitz (1999) suggest using 10 risk score groups. Nevertheless, based on simulation results, May and Hosmer (2004) show that, for all samples or samples with a large percentage of censored observations, the test rejects the null hypothesis too often. They suggest that the number of groups be chosen such that G=integer of {maximum of 12 and minimum of 10}. Lemeshow et al. (2004) state that the observations are firstly sorted in increasing order of their estimated event probability. / Thesis (M.Sc.)-University of KwaZulu-Natal, Pietermaritzburg, 2013.

A estatística é uma ciência com seus conceitos e métodos de coleta, organização e analise de informações que fazem parte dos currículos escolares da educação básica, na prática dos professores de matemática, para que os alunos compreendam, analisem e formem opinião crítica em relação às questões econômicas e sociais. A presente pesquisa buscou refletir sobre as práticas pedagógicas do professor de matemática no ensino de estatística no ensino médio, tendo como base as orientações para o ensino de estatísticas nas propostas dos Parâmetros Curriculares do Ensino Médio, as contribuições da aprendizagem significativa no ensino de estatística, com o uso das tecnologias na educação, através da proposta de planos de trabalho que abordem os conteúdos do ensino de estatística e a utilização do software livreCalc. Em relação aos caminhos metodológicos foi realizada uma pesquisa bibliográfica, utilizando o método de abordagem dedutivo, através de documentação indireta tendo como fonte de pesquisa os trabalhos científicos com foco no ensino e na aprendizagem da Estatística e da Probabilidade na Educação Básica. O desenvolvimento desta pesquisa possibilitou evidenciar caminhos metodológicos a serem desenvolvidos por professores de matemática na educação básica que contribuam na interpretação de dados do cotidiano a partir de análise de tabelas, análise de gráficos, medidas de posição, medidas de dispersão e linhas de tendência, utilizando como ferramentas as Tecnologias da Informação e Comunicação tendo como fundamentação teórica as contribuições de David Ausubel o conceito de aprendizagem significativa.

A estatística é uma ciência com seus conceitos e métodos de coleta, organização e analise de informações que fazem parte dos currículos escolares da educação básica, na prática dos professores de matemática, para que os alunos compreendam, analisem e formem opinião crítica em relação às questões econômicas e sociais. A presente pesquisa buscou refletir sobre as práticas pedagógicas do professor de matemática no ensino de estatística no ensino médio, tendo como base as orientações para o ensino de estatísticas nas propostas dos Parâmetros Curriculares do Ensino Médio, as contribuições da aprendizagem significativa no ensino de estatística, com o uso das tecnologias na educação, através da proposta de planos de trabalho que abordem os conteúdos do ensino de estatística e a utilização do software livreCalc. Em relação aos caminhos metodológicos foi realizada uma pesquisa bibliográfica, utilizando o método de abordagem dedutivo, através de documentação indireta tendo como fonte de pesquisa os trabalhos científicos com foco no ensino e na aprendizagem da Estatística e da Probabilidade na Educação Básica. O desenvolvimento desta pesquisa possibilitou evidenciar caminhos metodológicos a serem desenvolvidos por professores de matemática na educação básica que contribuam na interpretação de dados do cotidiano a partir de análise de tabelas, análise de gráficos, medidas de posição, medidas de dispersão e linhas de tendência, utilizando como ferramentas as Tecnologias da Informação e Comunicação tendo como fundamentação teórica as contribuições de David Ausubel o conceito de aprendizagem significativa.