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Neste trabalho são apresentadas algumas extensões dos modelos potência-alfa assumindo o contexto em que as observações estão censuradas ou limitadas. Inicialmente propomos um novo modelo assimétrico que estende os modelos t-assimétrico (Azzalini e Capitanio, 2003) e t-potência (Zhao e Kim, 2016) e inclui a distribuição t de Student como caso particular. Este novo modelo é capaz de ajustar dados com alto grau de assimetria e curtose, ainda maior do que os modelos t-assimétrico e t-potência. Em seguida estendemos o modelo t-potência às situações em que os dados apresentam censura, com alto grau de assimetria e caudas pesadas. Este modelo generaliza o modelo de regressão linear t de Student para dados censurados por Arellano-Valle et al. (2012). O trabalho também introduz o modelo linear misto normal-potência para dados assimétricos. Aqui a inferência estatística é realizada desde uma perspectiva clássica usando o método de máxima verossimilhança junto com o método de integração numérica de Gauss-Hermite para aproximar as integrais envolvidas na função de verossimilhança. Mais tarde, o modelo linear com interceptos aleatórios para dados duplamente censurados é estudado. Este modelo é desenvolvido sob a suposição de que os erros e os efeitos aleatórios seguem distribuições normal-potência e normal- assimétrica. Para todos os modelos estudados foram realizados estudos de simulação a fim de estudar as suas bondades de ajuste e limitações. Finalmente, ilustram-se todos os métodos propostos com dados reais. / In this work some extensions of the alpha-power models are presented, assuming the context in which the observations are censored or limited. Initially we propose a new asymmetric model that extends the skew-t (Azzalini e Capitanio, 2003) and power-t (Zhao e Kim, 2016) models and includes the Students t-distribution as a particular case. This new model is able to adjust data with a high degree of asymmetry and cursose, even higher than the skew-t and power-t models. Then we extend the power-t model to situations in which the data present censorship, with a high degree of asymmetry and heavy tails. This model generalizes the Students t linear censored regression model t by Arellano-Valle et al. (2012) The work also introduces the power-normal linear mixed model for asymmetric data. Here statistical inference is performed from a classical perspective using the maximum likelihood method together with the Gauss-Hermite numerical integration method to approximate the integrals involved in the likelihood function. Later, the linear model with random intercepts for doubly censored data is studied. This model is developed under the assumption that errors and random effects follow power-normal and skew-normal distributions. For all the models studied, simulation studies were carried out to study their benefits and limitations. Finally, all proposed methods with real data are illustrated.

Neste trabalho são apresentadas algumas extensões dos modelos potência-alfa assumindo o contexto em que as observações estão censuradas ou limitadas. Inicialmente propomos um novo modelo assimétrico que estende os modelos t-assimétrico (Azzalini e Capitanio, 2003) e t-potência (Zhao e Kim, 2016) e inclui a distribuição t de Student como caso particular. Este novo modelo é capaz de ajustar dados com alto grau de assimetria e curtose, ainda maior do que os modelos t-assimétrico e t-potência. Em seguida estendemos o modelo t-potência às situações em que os dados apresentam censura, com alto grau de assimetria e caudas pesadas. Este modelo generaliza o modelo de regressão linear t de Student para dados censurados por Arellano-Valle et al. (2012). O trabalho também introduz o modelo linear misto normal-potência para dados assimétricos. Aqui a inferência estatística é realizada desde uma perspectiva clássica usando o método de máxima verossimilhança junto com o método de integração numérica de Gauss-Hermite para aproximar as integrais envolvidas na função de verossimilhança. Mais tarde, o modelo linear com interceptos aleatórios para dados duplamente censurados é estudado. Este modelo é desenvolvido sob a suposição de que os erros e os efeitos aleatórios seguem distribuições normal-potência e normal- assimétrica. Para todos os modelos estudados foram realizados estudos de simulação a fim de estudar as suas bondades de ajuste e limitações. Finalmente, ilustram-se todos os métodos propostos com dados reais. / In this work some extensions of the alpha-power models are presented, assuming the context in which the observations are censored or limited. Initially we propose a new asymmetric model that extends the skew-t (Azzalini e Capitanio, 2003) and power-t (Zhao e Kim, 2016) models and includes the Students t-distribution as a particular case. This new model is able to adjust data with a high degree of asymmetry and cursose, even higher than the skew-t and power-t models. Then we extend the power-t model to situations in which the data present censorship, with a high degree of asymmetry and heavy tails. This model generalizes the Students t linear censored regression model t by Arellano-Valle et al. (2012) The work also introduces the power-normal linear mixed model for asymmetric data. Here statistical inference is performed from a classical perspective using the maximum likelihood method together with the Gauss-Hermite numerical integration method to approximate the integrals involved in the likelihood function. Later, the linear model with random intercepts for doubly censored data is studied. This model is developed under the assumption that errors and random effects follow power-normal and skew-normal distributions. For all the models studied, simulation studies were carried out to study their benefits and limitations. Finally, all proposed methods with real data are illustrated.

Made available in DSpace on 2016-06-02T20:06:02Z (GMT). No. of bitstreams: 1 2390.pdf: 3256865 bytes, checksum: 7ad1acbefc5f29dddbaad3f14dbcef7c (MD5) Previous issue date: 2009-04-17 / Financiadora de Estudos e Projetos / Microarrays technologies are used to measure the expression levels of a large amount of genes or fragments of genes simultaneously in diferent situations. This technology is useful to determine genes that are responsible for genetic diseases. A common statistical methodology used to determine whether a gene g has evidences to diferent expression levels is the t-test which requires the assumption of normality for the data (Saraiva, 2006; Baldi & Long, 2001). However this assumption sometimes does not agree with the nature of the analyzed data. In this work we use the skew-normal distribution described formally by Azzalini (1985), which has the normal distribution as a particular case, in order to relax the assumption of normality. Considering a frequentist approach we made a simulation study to detect diferences between the gene expression levels in situations of control and treatment through the t-test. Another simulation was made to examine the power of the t-test when we assume an asymmetrical model for the data. Also we used the likelihood ratio test to verify the adequability of an asymmetrical model for the data. / Os microarrays são ferramentas utilizadas para medir os níveis de expressão de uma grande quantidade de genes ou fragmentos de genes simultaneamente em situações variadas. Com esta ferramenta é possível determinar possíveis genes causadores de doenças de origem genética. Uma abordagem estatística comumente utilizada para determinar se um gene g apresenta evidências para níveis de expressão diferentes consiste no teste t, que exige a suposição de normalidade aos dados (Saraiva, 2006; Baldi & Long, 2001). No entanto, esta suposição pode não condizer com a natureza dos dados analisados. Neste trabalho, será utilizada a distribuição normal assimétrica descrita formalmente por Azzalini (1985), que tem a distribuição normal como caso particular, com o intuito de flexibilizar a suposição de normalidade. Considerando a abordagem clássica, é realizado um estudo de simulação para detectar diferenças entre os níveis de expressão gênica em situações de controle e tratamento através do teste t, também é considerado um estudo de simulação para analisar o poder do teste t quando é assumido um modelo assimétrico para o conjunto de dados. Também é realizado o teste da razão de verossimilhança, para verificar se o ajuste de um modelo assimétrico aos dados é adequado.

Submitted by Divisão de Documentação/BC Biblioteca Central (ddbc@ufam.edu.br) on 2018-05-29T14:38:33Z No. of bitstreams: 2 license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) VersaoFinal.pdf: 2882901 bytes, checksum: a35c6d27fe0f9aa61dfe3a96244b3140 (MD5) / Approved for entry into archive by Divisão de Documentação/BC Biblioteca Central (ddbc@ufam.edu.br) on 2018-05-29T14:38:46Z (GMT) No. of bitstreams: 2 license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) VersaoFinal.pdf: 2882901 bytes, checksum: a35c6d27fe0f9aa61dfe3a96244b3140 (MD5) / Made available in DSpace on 2018-05-29T14:38:46Z (GMT). No. of bitstreams: 2 license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) VersaoFinal.pdf: 2882901 bytes, checksum: a35c6d27fe0f9aa61dfe3a96244b3140 (MD5) Previous issue date: 2018-03-12 / CAPES - Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / The traditional estimation of mixture regression models is based on the assumption of normality of component errors and thus is sensitive to outliers, heavy-tailed and/or asymmetric errors. Another drawback is that, in general, the analysis is restricted to directly observed predictors. We present a proposal to deal with these issues simultaneously in the context of mixture regression by extending the classic normal model by assuming that, for each mixture component, the random errors and the covariates jointly follow a scale mixture of skew-normal distributions. It is also assumed that the covariates are observed with error. An MCMC-type algorithm to perform Bayesian inference is developed and, in order to show the efficacy of the proposed methods, simulated and real data sets are analyzed. / A estimação tradicional em mistura de modelos de regressão é baseada na suposição de normalidade para os erros aleatórios, sendo assim, sensível a outliers, caudas pesadas e erros assimétricos. Outra desvantagem é que, em geral, a análise é restrita a preditores que são observados diretamente. Apresentamos uma proposta para lidar com estas questões simultaneamente no contexto de mistura de regressões estendendo o modelo normal clássico. Assumimos que, conjuntamente e em cada componente da mistura, os erros aleatórios e as covariáveis seguem uma mistura de escala da distribuição normal assimétrica. Além disso, é feita a suposição de que as covariáveis são observadas com erro aditivo. Um algorítmo do tipo MCMC foi desenvolvido para realizar inferência Bayesiana. A eficácia do modelo proposto é verificada via análises de dados simulados e reais.

A Growth Curve Model (GCM) is a multivariate linear model used for analyzing longitudinal data with short to moderate time series. It is a special case of Generalized Multivariate Analysis of Variance (GMANOVA) models. Analysis using the GCM involves comparison of mean growths among different groups. The classical GCM, however, possesses some limitations including distributional assumptions, assumption of identical degree of polynomials for all groups and it requires larger sample size than the number of time points. In this thesis, we relax some of the assumptions of the traditional GCM and develop appropriate inferential tools for its analysis, with the aim of reducing bias, improving precision and to gain increased power as well as overcome limitations of high-dimensionality. Existing methods for estimating the parameters of the GCM assume that the underlying distribution for the error terms is multivariate normal. In practical problems, however, we often come across skewed data and hence estimation techniques developed under the normality assumption may not be optimal. Simulation studies conducted in this thesis, in fact, show that existing methods are sensitive to the presence of skewness in the data, where estimators are associated with increased bias and mean square error (MSE), when the normality assumption is violated. Methods appropriate for skewed distributions are, therefore, required. In this thesis, we relax the distributional assumption of the GCM and provide estimators for the mean and covariance matrices of the GCM under multivariate skew normal (MSN) distribution. An estimator for the additional skewness parameter of the MSN distribution is also provided. The estimators are derived using the expectation maximization (EM) algorithm and extensive simulations are performed to examine the performance of the estimators. Comparisons with existing estimators show that our estimators perform better than existing estimators, when the underlying distribution is multivariate skew normal. Illustration using real data set is also provided, wherein Triglyceride levels from the Framingham Heart Study is modelled over time. The GCM assumes equal degree of polynomial for each group. Therefore, when groups means follow different shapes of polynomials, the GCM fails to accommodate this difference in one model. We consider an extension of the GCM, wherein mean responses from different groups can have different shapes, represented by polynomials of different degree. Such a model is referred to as Extended Growth Curve Model (EGCM). We extend our work on GCM to EGCM, and develop estimators for the mean and covariance matrices under MSN errors. We adopted the Restricted Expectation Maximization (REM) algorithm, which is based on the multivariate Newton-Raphson (NR) method and Lagrangian optimization. However, the multivariate NR method and hence, the existing REM algorithm are applicable to vector parameters and the parameters of interest in this study are matrices. We, therefore, extended the NR approach to matrix parameters, which consequently allowed us to extend the REM algorithm to matrix parameters. The performance of the proposed estimators were examined using extensive simulations and a motivating real data example was provided to illustrate the application of the proposed estimators. Finally, this thesis deals with high-dimensional application of GCM. Existing methods for a GCM are developed under the assumption of ‘small p large n’ (n >> p) and are not appropriate for analyzing high-dimensional longitudinal data, due to singularity of the sample covariance matrix. In a previous work, we used Moore-Penrose generalized inverse to overcome this challenge. However, the method has some limitations around near singularity, when p~n. In this thesis, a Bayesian framework was used to derive a test for testing the linear hypothesis on the mean parameter of the GCM, which is applicable in high-dimensional situations. Extensive simulations are performed to investigate the performance of the test statistic and establish optimality characteristics. Results show that this test performs well, under different conditions, including the near singularity zone. Sensitivity of the test to mis-specification of the parameters of the prior distribution are also examined empirically. A numerical example is provided to illustrate the usefulness of the proposed method in practical situations. / Thesis / Doctor of Philosophy (PhD)

依據之前的文獻研究，市場上主要是在LHP (Large Homogeneous Portfolio) 假設下利用單因子常態關聯結構模式(One factor double Gaussian copula model) 評價擔保債權憑證 (Collateralized debt obligation, CDO)。但這會造成擔保債權憑證的評價與市場報價的差距過大，且會造成base correlation偏斜的情況。Kalemanova et al. (2007) 提出用Normal inverse Gaussian (NIG) 取代常態分配評價擔保債權憑證，此模型不但計算快速而且可以準確估計權益分券 (equity tranche) 的價格，但是它也過於高估了其它的分券的價格。 在本文中使用多變量封閉常態分配(Closed skew normal, 簡稱CSN) 分配取代NIG分配作擔保債權憑證分券的評價，CSN分配具有常態分配的性質，其線性組合仍具有封閉性的特質，且具有較多的參數以控制分配的偏態與峰態。但是與單因子常態關聯結構模式相同，多變量封閉常態分配的單因子關聯結構模式仍然無法估計的很準確，僅有在最高等級分券(senior tranche)的評價上有明顯的改進。 因此在本文中我們使用NIG與CSN複合分配之單因子關聯結構模式評價擔保債權憑證分券，在實例分析時得到極佳的評價結果，並且比單因子常態關聯結構模型具有更多的的參數以使模型更符合實際的需求。 / This article extends the Large Homogeneous Portfolio (LHP) and one factor double Gaussian copula approach for pricing CDOs. In the literature, the one factor double Gaussian copula model under LHP assumption fails to fit the prices of CDO tranches, moreover, it leads to the implied base correlation skew. Some researchers proposed using one factor double NIG copula model to price CDO tranches. It not only economizes on time but also fits the equity tranches exactly, but NIG models do not price other tranches well simultaneously. On the other hand, we substitute the NIG distribution with the Closed Skew normal (CSN) distribution. This family also has properties similar to the normal distribution, which is closure under convolution, and has extra parameters to control the shape. By using this model we get a better fit in the senior tranches, but it seriously overprices subordinate tranches. Thus we consider a mixture distribution of NIG and CSN distributions. The employments of this mixture distribution are comparatively well, and furthermore it brings more flexibility to the dependence structure.

The aim of this work is to present an inference and diagnostic study of an extension of the lifetime distribution family proposed by Birnbaum and Saunders (1969a,b). This extension is obtained by considering a skew-elliptical distribution instead of the normal distribution. Specifically, in this work we develop a Birnbaum-Saunders (BS) distribution type based on scale mixtures of skew-normal distributions (SMSN). The resulting family of lifetime distributions represents a robust extension of the usual BS distribution. Based on this family, we reproduce the usual properties of the BS distribution, and present an estimation method based on the EM algorithm. In addition, we present regression models associated with the BS distributions (based on scale mixtures of skew-normal), which are developed as an extension of the sinh-normal distribution (Rieck and Nedelman, 1991). For this model we consider an estimation and diagnostic study for uncensored data. / O objetivo deste trabalho é apresentar um estudo de inferência e diagnóstico em uma extensão da família de distribuições de tempos de vida proposta por Birnbaum e Saunders (1969a,b). Esta extensão é obtida ao considerar uma distribuição skew-elíptica em lugar da distribuição normal. Especificamente, neste trabalho desenvolveremos um tipo de distribuição Birnbaum-Saunders (BS) baseda nas distribuições mistura de escala skew-normal (MESN). Esta família resultante de distribuições de tempos de vida representa uma extensão robusta da distribuição BS usual. Baseado nesta família, vamos reproduzir as propriedades usuais da distribuição BS, e apresentar um método de estimação baseado no algoritmo EM. Além disso, vamos apresentar modelos de regressão associado à distribuições BS (baseada na distribuição mistura de escala skew-normal), que é desenvolvida como uma extensão da distribuição senh-normal (Rieck e Nedelman, 1991), para estes vamos considerar um estudo de estimação e diagnóstisco para dados sem censura.

The aim of this work is to present an inference and diagnostic study of an extension of the lifetime distribution family proposed by Birnbaum and Saunders (1969a,b). This extension is obtained by considering a skew-elliptical distribution instead of the normal distribution. Specifically, in this work we develop a Birnbaum-Saunders (BS) distribution type based on scale mixtures of skew-normal distributions (SMSN). The resulting family of lifetime distributions represents a robust extension of the usual BS distribution. Based on this family, we reproduce the usual properties of the BS distribution, and present an estimation method based on the EM algorithm. In addition, we present regression models associated with the BS distributions (based on scale mixtures of skew-normal), which are developed as an extension of the sinh-normal distribution (Rieck and Nedelman, 1991). For this model we consider an estimation and diagnostic study for uncensored data. / O objetivo deste trabalho é apresentar um estudo de inferência e diagnóstico em uma extensão da família de distribuições de tempos de vida proposta por Birnbaum e Saunders (1969a,b). Esta extensão é obtida ao considerar uma distribuição skew-elíptica em lugar da distribuição normal. Especificamente, neste trabalho desenvolveremos um tipo de distribuição Birnbaum-Saunders (BS) baseda nas distribuições mistura de escala skew-normal (MESN). Esta família resultante de distribuições de tempos de vida representa uma extensão robusta da distribuição BS usual. Baseado nesta família, vamos reproduzir as propriedades usuais da distribuição BS, e apresentar um método de estimação baseado no algoritmo EM. Além disso, vamos apresentar modelos de regressão associado à distribuições BS (baseada na distribuição mistura de escala skew-normal), que é desenvolvida como uma extensão da distribuição senh-normal (Rieck e Nedelman, 1991), para estes vamos considerar um estudo de estimação e diagnóstisco para dados sem censura.

This thesis deals with several statistical and probabilistic aspects of symmetry and asymmetry, both in a univariate and multivariate context, and is divided into three distinct parts. The first part, composed of Chapters 1, 2 and 3 of the thesis, solves two conjectures associated with multivariate skew-symmetric distributions. Since the introduction in 1985 by Adelchi Azzalini of the most famous representative of that class of distributions, namely the skew-normal distribution, it is well-known that, in the vicinity of symmetry, the Fisher information matrix is singular and the profile log-likelihood function for skewness admits a stationary point whatever the sample under consideration. Since that moment, researchers have tried to determine the subclasses of skew-symmetric distributions who suffer from each of those problems, which has led to the aforementioned two conjectures. This thesis completely solves these two problems. The second part of the thesis, namely Chapters 4 and 5, aims at applying and constructing extremely general skewing mechanisms. As such, in Chapter 4, we make use of the univariate mechanism of Ferreira and Steel (2006) to build optimal (in the Le Cam sense) tests for univariate symmetry which are very flexible. Actually, their mechanism allowing to turn a given symmetric distribution into any asymmetric distribution, the alternatives to the null hypothesis of symmetry can take any possible shape. These univariate mechanisms, besides that surjectivity property, enjoy numerous good properties, but cannot be extended to higher dimensions in a satisfactory way. For this reason, we propose in Chapter 5 different general mechanisms, sharing all the nice properties of their competitors in Ferreira and Steel (2006), but which moreover can be extended to any dimension. We formally prove that the surjectivity property holds in dimensions k>1 and we study the principal characteristics of these new multivariate mechanisms. Finally, the third part of this thesis, composed of Chapter 6, proposes a test for multivariate central symmetry by having recourse to the concepts of statistical depth and runs. This test extends the celebrated univariate runs test of McWilliams (1990) to higher dimensions. We analyze its asymptotic behavior (especially in dimension k=2) under the null hypothesis and its invariance and robustness properties. We conclude by an overview of possible modifications of these new tests./ Cette thèse traite de différents aspects statistiques et probabilistes de symétrie et asymétrie univariées et multivariées, et est subdivisée en trois parties distinctes. La première partie, qui comprend les chapitres 1, 2 et 3 de la thèse, est destinée à la résolution de deux conjectures associées aux lois skew-symétriques multivariées. Depuis l'introduction en 1985 par Adelchi Azzalini du plus célèbre représentant de cette classe de lois, à savoir la loi skew-normale, il est bien connu qu'en un voisinage de la situation symétrique la matrice d'information de Fisher est singulière et la fonction de vraisemblance profile pour le paramètre d'asymétrie admet un point stationnaire quel que soit l'échantillon considéré. Dès lors, des chercheurs ont essayé de déterminer les sous-classes de lois skew-symétriques qui souffrent de chacune de ces problématiques, ce qui a mené aux deux conjectures précitées. Cette thèse résoud complètement ces deux problèmes. La deuxième partie, constituée des chapitres 4 et 5, poursuit le but d'appliquer et de proposer des méchanismes d'asymétrisation très généraux. Ainsi, au chapitre 4, nous utilisons le méchanisme univarié de Ferreira and Steel (2006) pour construire des tests de symétrie univariée optimaux (au sens de Le Cam) qui sont très flexibles. En effet, leur méchanisme permettant de transformer une loi symétrique donnée en n'importe quelle loi asymétrique, les contre-hypothèses à la symétrie peuvent prendre toute forme imaginable. Ces méchanismes univariés, outre cette propriété de surjectivité, possèdent de nombreux autres attraits, mais ne permettent pas une extension satisfaisante aux dimensions supérieures. Pour cette raison, nous proposons au chapitre 5 des méchanismes généraux alternatifs, qui partagent toutes les propriétés de leurs compétiteurs de Ferreira and Steel (2006), mais qui en plus sont généralisables à n'importe quelle dimension. Nous démontrons formellement que la surjectivité tient en dimension k > 1 et étudions les caractéristiques principales de ces nouveaux méchanismes multivariés. Finalement, la troisième partie de cette thèse, composée du chapitre 6, propose un test de symétrie centrale multivariée en ayant recours aux concepts de profondeur statistique et de runs. Ce test étend le célèbre test de runs univarié de McWilliams (1990) aux dimensions supérieures. Nous en analysons le comportement asymptotique (surtout en dimension k = 2) sous l'hypothèse nulle et les propriétés d'invariance et de robustesse. Nous concluons par un aperçu sur des modifications possibles de ces nouveaux tests.