<p>When researchers work on time series or sequence, certain fundamental questions will naturally arise. One of them will be whether the series or sequence exhibits a gradual trend over time. In this thesis, we propose a test statistic based on moving order statistics and establish an exact procedure to test for the presence of monotone trends. We show that the test statistic under the null hypothesis that there is no trend follows the closed skew normal distribution. An efficient algorithm is then developed to generate realizations from this null distribution. A simulation study is conducted to evaluate the proposed test under the alternative hypotheses with linear, logarithmic and quadratic trend functions. Finally, a practical example is provided to illustrate the proposed test procedure.</p> / Master of Science (MSc)

In this thesis we introduce, implement and compare several multivariate goodness-of-fit tests. First of all, we will focus on universal mul- tivariate tests that do not place any assumptions on parametric families of null distributions. Thereafter, we will be concerned with testing of multi- variate normality and, by using Monte Carlo simulations, we will compare power of five different tests of bivariate normality against several alternati- ves. Then we describe multivariate skew-normal distribution and propose a new test of multivariate skew-normality based on empirical moment genera- ting functions. In the final analysis, we compare its power with other tests of multivariate skew-normality. 1

In the present work we study families of skew-probability distributions. We will gradually build concept of families of more and more general distributions. For us the most important ones are skew normal distribution, elliptical distri- bution and skew elliptical distribution. On the each of them we will present basic properties and visualize particular examples. At the end we will generate realizations of variates and propose how to estimate the original distribution.

In 2007, Domínguez-Molina et al. obtained the moment generating function (mgf) of the matrix variate closed skew normal distribution. In this paper, we use their mgf to obtain the first two moments and some additional properties of quadratic forms for the matrix variate skew normal distributions. The quadratic forms are particularly interesting because they are essentially correlation tests that introduce a new type of orthogonality condition.

Matrix variate skew elliptically contoured distributions generalize several classes of important distributions. This paper defines and explores matrix variate skew elliptically contoured distributions. In particular, we discuss the first two moments of the matrix variate skew elliptically contoured distributions.

The study of properties of probability distributions has always been a persistent theme of statistics and of applied probability. This thesis deals with an investigation of distribution functions under the following two topics: (i) characterization of distributions based on record values and order statistics, (ii) properties of the skew-t distribution. Within the extensive characterization literature there are several results involving properties of record values and order statistics. Although there have been many well known results already developed, it is still of great interest to find new characterization of distributions based on record values and order statistics. In the first part, we provide the conditional distribution of any record value given the maximum order statistics and study characterizations of distributions based on record values and the maximum order statistics. We also give some characterizations of the mean value function within the class of order statistics point processes, by using certain relations between the conditional moments of the jump times or current lives. These results can be applied to characterize the uniform distribution using the sequence of order statistics, and the exponential distribution using the sequence of record values, respectively. Azzalini (1985, 1986) introduced the skew-normal distribution which includes the normal distribution and has some properties like the normal and yet is skew. This class of distributions is useful in studying robustness and for modeling skewness. Since then, skew-symmetric distributions have been proposed by many authors. In the second part, the so-called generalized skew-t distribution is defined and studied. Examples of distributions in this class, generated by the ratio of two independent skew-symmetric distributions, are given. We also investigate properties of the skew-symmetric distribution.

Orientador: Victor Hugo Lachos Davila / Dissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Matematica, Estatistica e Computação Cientifica / Made available in DSpace on 2018-08-13T07:03:11Z (GMT). No. of bitstreams: 1 Basso_RodrigoMarreiro_M.pdf: 3130269 bytes, checksum: 85e95beb812a4ec069f39f8b9c79681a (MD5) Previous issue date: 2009 / Resumo: Nesse trabalho será considerada uma classe flexível de modelos usando misturas finitas de distribuições da classe de misturas de escala skew-normal. O algoritmo EM é empregado para se obter estimativas de máxima verossimilhança de maneira iterativa, sendo discutido com maior ênfase para misturas de distribuições skew-normal, skew-t, skew-slash e skew-normal contaminada. Também será apresentado um método geral para aproximar a matrix de covariância assintótica das estimativas de máxima verossimilhança. Resultados obtidos da análise de quatro conjuntos de dados reais ilustram a aplicabilidade da metodologia proposta / Abstract: In this work we consider a flexible class of models using finite mixtures of multivariate scale mixtures of skew-normal distributions. An EM-type algorithm is employed for iteratively computing maximum likelihood estimates and this is discussed with emphasis on finite mixtures of skew-normal, skew-t, skew-slash and skew-contaminated normal distributions. A general information-based method for approximating the asymptotic covariance matrix of the maximum likelihood estimates is also presented. Results obtained from the analysis of four real data sets are reported illustrating the usefulness of the proposed methodology / Mestrado / Mestre em Estatística

Orientador: Caio Lucidius Naberezny Azevedo / Dissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Matemática Estatística e Computação Científica / Made available in DSpace on 2018-08-24T12:51:18Z (GMT). No. of bitstreams: 1 Ferreira_EduardoVargas_M.pdf: 8131052 bytes, checksum: f344cd1f11e8d818f3aac90f48396cbc (MD5) Previous issue date: 2014 / Resumo: No presente trabalho propõem-se novos modelos da Teoria de Resposta ao Item para respostas politômicas nominais e ordinais (graduais), via dados aumentados, para grupos múltiplos. Para a modelagem das distribuições dos traços latentes de cada grupo, considera-se normais assimétricas centradas. Tal abordagem, além de acomodar a característica de assimetria aos dados, ajuda a garantir a identificabilidade dos modelos estudados, a qual é tratada tanto sob a ótica frequentista quanto bayesiana. Com relação aos métodos de estimação, desenvolveu-se procedimentos bayesianos através de algoritmos de Monte Carlo via cadeias de Markov (MCMC), utilizando o algoritmo de Gibbs (DAGS), com a verossimilhança aumentada (dados aumentados) e Metropolis-Hastings, considerando a verossimilhança original. As implementações computacionais foram escritas em linguagem C++, integradas ao ambiente computacional, gráfico e estatístico R, viabilizando rotinas gratuitas, de código aberto e alta velocidade no processamento, essenciais à difusão de tais metodologias. Para a seleção de modelos, utilizou-se o critério de informação deviance (DIC), os valores esperados do critério de informação de Akaike (EAIC) e o critério de informação bayesiano (EBIC). Em relação à verificação da qualidade do ajuste de modelos, explorou-se a checagem preditiva a posteriori, que fornece meios concretos de se avaliar a qualidade do instrumento de medida (prova, questionário etc), qualidade do ajuste do modelo de um modo global, além de indícios de violações de suposições específicas. Estudos de simulação, considerando diversas situações de interesse prático, indicam que os modelos e métodos de estimação produzem resultados bastante satisfatórios, com superioridade dos modelos assimétricos com relação ao simétrico (o qual assume simetria das distribuições das variáveis latentes). A análise de um conjunto de dados reais, referente à primeira fase do vestibular da UNICAMP de 2013, ilustra o potencial da tríade: modelagem, métodos de estimação e ferramentas de diagnósticos, desenvolvida neste trabalho / Abstract: In this work, we propose new Item Response Theory models for nominal and ordinal (gradual) polytomous responses through augmented data schemes considering multiple groups. For the distribution of the latent traits of each group, we consider a skew-normal distribution under the centered parametrization. This approach will allow for accommodating a possible skewness of the latent trait distribution, but is also helpful to ensure the identifiability of the models, which is studied under frequentist and Bayesian paradigms. Concerning estimation methods, we developed Bayesian methods through Markov chain Monte Carlo (MCMC) algorithms by using the Gibbs algorithm (DAGS), with augmented likelihood (augmented data) and Metropolis-Hastings algorithms, considering the original likelihood. The computational environment was written in the C++ language and integrated with the R program (a statistical computational and graphical environment), allowing for free, open source and high-speed routines which, in turn, are essential to the dissemination of the developed methodologies. In terms of model selection, we considered the deviance information criterion (DIC), the expected Akaike information criterion (EAIC) and expected Bayesian information criterion (EBIC). Regarding model-fit assessment tools, we explore the posterior predictive model- checking which allows for assessing the quality of measurement, instruments (tests, questionnaires, and others), the model fit in a global sense, besides providing directions toward violations of specific assumptions. Simulation studies, considering different situations of practical interest, indicate that the models and estimation methods produced reasonable results, with outperformance of skew models when compared to symmetric ones (which assumes symmetry of the latent trait distribution). Analysis of a data set which corresponds to the first phase of the 2013 written examination of UNICAMP (State University of Campinas), illustrates the potential of the following triad: modelling; estimation methods; and diagnostic tools developed in this work. / Mestrado / Estatistica / Mestre em Estatística

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