The consequences of substituting the denominator Q 3(p) - Q 1(p) by Q 2 - Q 1(p) in Groeneveld's class of quantile measures of kurtosis (γ 2(p)) for symmetric distributions, are explored using the symmetric influence function. The relationship between the measure γ 2(p) and the alternative class of kurtosis measures κ2(p) is derived together with the relationship between their influence functions. The Laplace, Logistic, symmetric Two-sided Power, Tukey and Beta distributions are considered in the examples in order to discuss the results obtained pertaining to unimodal, heavy tailed, bounded domain and U-shaped distributions.

Submitted by Marlene Santos (marlene.bc.ufg@gmail.com) on 2017-09-22T19:42:42Z No. of bitstreams: 2 Dissertação - Leandro Valerio Silva - 2017.pdf: 5149451 bytes, checksum: 3c193b0556f60d79c470ef940136aaf7 (MD5) license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) / Approved for entry into archive by Luciana Ferreira (lucgeral@gmail.com) on 2017-09-25T11:46:04Z (GMT) No. of bitstreams: 2 Dissertação - Leandro Valerio Silva - 2017.pdf: 5149451 bytes, checksum: 3c193b0556f60d79c470ef940136aaf7 (MD5) license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) / Made available in DSpace on 2017-09-25T11:46:04Z (GMT). No. of bitstreams: 2 Dissertação - Leandro Valerio Silva - 2017.pdf: 5149451 bytes, checksum: 3c193b0556f60d79c470ef940136aaf7 (MD5) license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) Previous issue date: 2017-03-25 / The aim of this study is to use control charts to monitor the ICMS revenue in Goias, Brazil. The autocorrelation of the data was removed by using symmetric autoregressive moving average (SYMARMA) models. The results showed that the Shewhart control chart obtained from the SYMARMA model, based on a conditional normal distribuition, presented the best result. / O objetivo principal deste estudo é construir gráficos de controle para o monitoramento da arrecadação de ICMS em Goiás. Para tal, foram utilizados modelos autoregressivos de médias móveis simétricos (SYMARMA) com a finalidade de eliminar a autocorrelação presente na série temporal dos recolhimentos de ICMS. Construíram-se gráficos de controle do tipo Shewhart, CUSUM e EWMA, a partir dos resíduos gerados pelo modelo SYMARMA. Os resultados demonstraram que o gráfico de controle de Shewhart construído a partir dos resíduos do modelo SYMARMA com distribuição normal condicional apresentou-se o mais de acordo com a realidade da arrecadação.

Neste trabalho foi proposto o modelo linear parcial generalizado simétrico, com base nos modelos lineares parciais generalizados e nos modelos lineares simétricos, em que a variável resposta segue uma distribuição que pertence à família de distribuições simétricas, considerando um preditor linear que possui uma parte paramétrica e uma não paramétrica. Algumas distribuições que pertencem a essa classe são as distribuições: Normal, t-Student, Exponencial potência, Slash e Hiperbólica, dentre outras. Uma breve revisão dos conceitos utilizados ao longo do trabalho foram apresentados, a saber: análise residual, influência local, parâmetro de suavização, spline, spline cúbico, spline cúbico natural e algoritmo backfitting, dentre outros. Além disso, é apresentada uma breve teoria dos modelos GAMLSS (modelos aditivos generalizados para posição, escala e forma). Os modelos foram ajustados utilizando o pacote gamlss disponível no software livre R. A seleção de modelos foi baseada no critério de Akaike (AIC). Finalmente, uma aplicação é apresentada com base em um conjunto de dados reais da área financeira do Chile. / In this work we propose the symmetric generalized partial linear model, based on the generalized partial linear models and symmetric linear models, that is, the response variable follows a distribution that belongs to the symmetric distribution family, considering a linear predictor that has a parametric and a non-parametric component. Some distributions that belong to this class are distributions: Normal, t-Student, Power Exponential, Slash and Hyperbolic among others. A brief review of the concepts used throughout the work was presented, namely: residual analysis, local influence, smoothing parameter, spline, cubic spline, natural cubic spline and backfitting algorithm, among others. In addition, a brief theory of GAMLSS models is presented (generalized additive models for position, scale and shape). The models were adjusted using the package gamlss available in the free R software. The model selection was based on the Akaike criterion (AIC). Finally, an application is presented based on a set of real data from Chile\'s financial area.

Neste trabalho foi proposto o modelo linear parcial generalizado simétrico, com base nos modelos lineares parciais generalizados e nos modelos lineares simétricos, em que a variável resposta segue uma distribuição que pertence à família de distribuições simétricas, considerando um preditor linear que possui uma parte paramétrica e uma não paramétrica. Algumas distribuições que pertencem a essa classe são as distribuições: Normal, t-Student, Exponencial potência, Slash e Hiperbólica, dentre outras. Uma breve revisão dos conceitos utilizados ao longo do trabalho foram apresentados, a saber: análise residual, influência local, parâmetro de suavização, spline, spline cúbico, spline cúbico natural e algoritmo backfitting, dentre outros. Além disso, é apresentada uma breve teoria dos modelos GAMLSS (modelos aditivos generalizados para posição, escala e forma). Os modelos foram ajustados utilizando o pacote gamlss disponível no software livre R. A seleção de modelos foi baseada no critério de Akaike (AIC). Finalmente, uma aplicação é apresentada com base em um conjunto de dados reais da área financeira do Chile. / In this work we propose the symmetric generalized partial linear model, based on the generalized partial linear models and symmetric linear models, that is, the response variable follows a distribution that belongs to the symmetric distribution family, considering a linear predictor that has a parametric and a non-parametric component. Some distributions that belong to this class are distributions: Normal, t-Student, Power Exponential, Slash and Hyperbolic among others. A brief review of the concepts used throughout the work was presented, namely: residual analysis, local influence, smoothing parameter, spline, cubic spline, natural cubic spline and backfitting algorithm, among others. In addition, a brief theory of GAMLSS models is presented (generalized additive models for position, scale and shape). The models were adjusted using the package gamlss available in the free R software. The model selection was based on the Akaike criterion (AIC). Finally, an application is presented based on a set of real data from Chile\'s financial area.

Submitted by (ana.araujo@ufrpe.br) on 2016-08-03T15:37:53Z No. of bitstreams: 1 Luiz Medeiros de Araujo Lima Filho.pdf: 6714156 bytes, checksum: 8537b197237b8261b5b264ad769ab466 (MD5) / Made available in DSpace on 2016-08-03T15:37:53Z (GMT). No. of bitstreams: 1 Luiz Medeiros de Araujo Lima Filho.pdf: 6714156 bytes, checksum: 8537b197237b8261b5b264ad769ab466 (MD5) Previous issue date: 2012-10-01 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / Knowledge of growth and production present and future of trees in forest stands is a key element in sustainable forest management. It is intended in this thesis to estimate the frequency distribution by diameter class, estimate and compare volumetric equations via symmetric nonlinear models as well, propose to compare models and adjust volume growth via symmetrical nonlinear models in Eucalyptus spp. clones in the region of the Araripe - PE. The experiment was conducted at the Experimental Station of the Agricultural Research Company of Pernambuco (IPA), located in the municipality of Araripina - PE. This experiment consists of 15 clones of Eucalyptus spp., species and hybrids resulting from natural crossing and controlled pollination. The database is composed of all the survival trees of the experiment, corresponding to 1189 trees, divided into three groups. To estimate the diameter distributions were adopted probability distributions Normal, with three parameters Weibull, Log-normal and Gamma. The next step was to fit of the model of Schumacher and Hall and Spurr model associated the following symmetric distributions: Normal, Student t, Exponential Power and Logistics II. The volume growth models in this thesis were adjusted with symmetric distributions associated with them. In the estimation of the diameter distributions, the results were similar in the three groups, with emphasis on the Log-normal distribution. This distribution was the most appropriate for most of the period. Analyzing the volumetric equations, it was found that the model was Schumacher and Hall the most suitable, when associated with the Student t distribution with three degrees .of freedom and Exponential Power (t = 0,8) respectively to groups I and II. In group III, the model Spurr associated with Exponential Power (t = 0,8) distribution had the best performance. Comparing the volume growth models, it was observed that the proposed models best fits obtained in this thesis, when the distributions associated with the Student t with three degrees of freedom and Exponential Power (t = 0,8), respectively, in groups I and III. In group II, the Chapman-Richards model associated with Student t distribution with three degrees of freedom had the best performance. / O conhecimento do crescimento e da produção presente e futura de árvores em povoamentos florestais é elemento fundamental no manejo florestal sustentável. Desta forma, pretende-se nesta tese estimar a distribuição de frequência por classe diamétrica, estimar e comparar equações volumétricas via modelos simétricos não lineares, bem como, propor, ajustar e comparar modelos de crescimento em volume via modelos simétricos não lineares em clones de Eucalyptus spp. na região da Chapada do Araripe - PE. O experimento foi desenvolvido na Estação Experimental da Empresa Pernambucana de Pesquisa Agropecuária (IPA), localizada no município de Araripina - PE. Esse experimento é composto por 15 clones de Eucalyptus spp., resultantes de espécies e híbridos de cruzamento natural e polinização controlada. A base de dados foi composta por todas as árvores sobreviventes do experimento, que corresponde a 1189 árvores, dividida em três grupos. Para estimar as distribuições diamétricas foram adotadas as distribuições de probabilidade Normal, Weibull com três parâmetros, Log-normal e Gama. Em seguida, procedeu-se com os ajustes dos modelos volumétricos de Schumacher e Hall e de Spurr associados as seguintes distribuições simétricos: Normal, t de Student, Exponencial Potência e Logística II. Posteriormente, os modelos de crescimento em volume propostos nesta tese foram ajustados associados às mesmas distribuições simétricas. Na estimativa das distribuições diamétricas, os resultados foram similares nos três grupos, com destaque para a distribuição Log-normal. Essa distribuição foi a mais significativa na maior parte do período avaliado. Analisando as equações volumétricas, observou-se que o modelo de Schumacher e Hall obteve o melhor desempenho, quando associado as distribuições t de Student com três graus de liberdade e Exponencial Potência (t = 0,8), respectivamente, para os grupos I e II. No grupo III, o modelo de Spurr associado a distribuição Exponencial Potência (t = 0,8) obteve o melhor desempenho. Comparando os modelos de crescimento em volume, observou-se que os modelos propostos nesta tese obtiveram melhores ajustes, quando associados as distribuições t de Student com três graus de liberdade e Exponencial Potência (t = 0,8), respectivamente, nos grupos I e III. No grupo II, o modelo de Chapman-Richards associado a distribuição t de Student com três graus de liberdade obteve o melhor desempenho.

Submitted by Franciele Moreira (francielemoreyra@gmail.com) on 2018-03-29T12:30:01Z No. of bitstreams: 2 Dissertação - Marcelo dos Santos Ventura - 2018.pdf: 4739813 bytes, checksum: 52211670f6e17c893ffd08843056f075 (MD5) license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) / Approved for entry into archive by Luciana Ferreira (lucgeral@gmail.com) on 2018-03-29T13:40:08Z (GMT) No. of bitstreams: 2 Dissertação - Marcelo dos Santos Ventura - 2018.pdf: 4739813 bytes, checksum: 52211670f6e17c893ffd08843056f075 (MD5) license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) / Made available in DSpace on 2018-03-29T13:40:08Z (GMT). No. of bitstreams: 2 Dissertação - Marcelo dos Santos Ventura - 2018.pdf: 4739813 bytes, checksum: 52211670f6e17c893ffd08843056f075 (MD5) license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) Previous issue date: 2018-03-09 / Fundação de Amparo à Pesquisa do Estado de Goiás - FAPEG / This work deals with two Monte Carlo simulation studies in log-symmetric regression models, which are particularly useful for the cases when the response variable is continuous, strictly positive and asymmetric, with the possibility of the existence of atypical observations. In log- symmetric regression models, the distribution of the random errors multiplicative belongs to the log-symmetric class, which encompasses log-normal, log- Student-t, log-power- exponential, log-slash, log-hyperbolic distributions, among others. The first simulation study has as objective to examine the performance for the maximum-likelihood estimators of the model parameters, where various scenarios are considered. The objective of the second simulation study is to investigate the accuracy of popular information criteria as AIC, BIC, HQIC and their respective corrected versions. As illustration, a movie data set obtained and assembled for this dissertation is analyzed to compare log-symmetric models with the normal linear model and to obtain the best model by using the mentioned information criteria. / Este trabalho aborda dois estudos de simulação de Monte Carlo em modelos de regressão log- simétricos, os quais são particularmente úteis para os casos em que a variável resposta é contínua, estritamente positiva e assimétrica, com possibilidade da existência de observações atípicas. Nos modelos de regressão log-simétricos, a distribuição dos erros aleatórios multiplicativos pertence à classe log-simétrica, a qual engloba as distribuições log-normal, log-Student- t, log-exponencial- potência, log-slash, log-hyperbólica, entre outras. O primeiro estudo de simulação tem como objetivo examinar o desempenho dos estimadores de máxima verossimilhança desses modelos, onde vários cenários são considerados. No segundo estudo de simulação o objetivo é investigar a eficácia critérios de informação populares como AIC, BIC, HQIC e suas respectivas versões corrigidas. Como ilustração, um conjunto de dados de filmes obtido e montado para essa dissertação é analisado para comparar os modelos de regressão log-simétricos com o modelo linear normal e para obter o melhor modelo utilizando os critérios de informação mencionados.

Beaucoup de résultats sur l’estimation d’une matrice d’échelle en analyse multidimensionnelle sont obtenus sous l’hypothèse de normalité (condition sous laquelle il s’agit de la matrice de covariance). Or il s’avère que, dans des domaines tels que la gestion de portefeuille en finance, cette hypothèse n’est pas très appropriée. Dans ce cas, la famille des distributions à symétrie elliptique, qui contient la distribution gaussienne, est une alternative intéressante. Nous considérons dans cette thèse le problème d’estimation de la matrice d’échelle Σ du modèle additif Yp_m = M + E, d’un point de vue de la théorie de la décision. Ici, p représente le nombre de variables, m le nombre d’observations, M une matrice de paramètres inconnus de rang q < p et E un bruit aléatoire de distribution à symétrie elliptique, avec une matrice de covariance proportionnelle à Im x Σ. Ce problème d’estimation est abordé sous la représentation canonique de ce modèle où la matrice d’observation Y est décomposée en deux matrices, à savoir, Zq x p qui résume l’information contenue dans M et une matrice Un x p, où n = m - q, qui résume l’information suffisante pour l’estimation de Σ. Comme les estimateurs naturels de la forme Σa = a S (où S = UT U et a est une constante positive) ne sont pas de bons estimateurs lorsque le nombre de variables p et le rapport p=n sont grands, nous proposons des estimateurs alternatifs de la forme ^Σa;G = a(S + S S+G(Z; S)) où S+ est l’inverse de Moore-Penrose de S (qui coïncide avec l’inverse S-1 lorsque S est inversible). Nous fournissons des conditions sur la matrice de correction SS+G(Z; S) telles que ^Σa;G améliore^Σa sous le coût quadratique L(Σ; ^Σ) = tr(^ΣΣ‾1 - Ip)² et sous une modification de ce dernier, à savoir le coût basé sur les données LS (Σ; ^Σ) = tr(S+Σ(^ΣΣ‾1 - Ip)²). Nous adoptons une approche unifiée des deux cas où S est inversible et S est non inversible. À cette fin, une nouvelle identité de type Stein-Haff et un nouveau calcul sur la décomposition en valeurs propres de S sont développés. Notre théorie est illustrée par une grande classe d’estimateurs orthogonalement invariants et par un ensemble de simulations. / Numerous results on the estimation of a scale matrix in multivariate analysis are obtained under Gaussian assumption (condition under which it is the covariance matrix). However in such areas as Portfolio management in finance, this assumption is not well adapted. Thus, the family of elliptical symmetric distribution, which contains the Gaussian distribution, is an interesting alternative. In this thesis, we consider the problem of estimating the scale matrix _ of the additif model Yp_m = M + E, under theoretical decision point of view. Here, p is the number of variables, m is the number of observations, M is a matrix of unknown parameters with rank q < p and E is a random noise, whose distribution is elliptically symmetric with covariance matrix proportional to Im x Σ. It is more convenient to deal with the canonical forme of this model where Y is decomposed in two matrices, namely, Zq_p which summarizes the information contained in M, and Un_p, where n = m - q which summarizes the information sufficient to estimate Σ. As the natural estimators of the form ^Σ a = a S (where S = UT U and a is a positive constant) perform poorly when the dimension of variables p and the ratio p=n are large, we propose estimators of the form ^Σa;G = a(S + S S+G(Z; S)) where S+ is the Moore-Penrose inverse of S (which coincides with S-1 when S is invertible). We provide conditions on the correction matrix SS+G(Z; S) such that ^Σa;G improves over ^Σa under the quadratic loss L(Σ; ^Σ) = tr(^ΣΣ‾1 - Ip)² and under the data based loss LS (Σ; ^Σ) = tr(S+Σ(^ΣΣ‾1 - Ip)²).. We adopt a unified approach of the two cases where S is invertible and S is non-invertible. To this end, a new Stein-Haff type identity and calculus on eigenstructure for S are developed. Our theory is illustrated with the large class of orthogonally invariant estimators and with simulations.

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.

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.<p><p>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.<p><p>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.<p><p>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./<p><p>Cette thèse traite de différents aspects statistiques et probabilistes de symétrie et asymétrie univariées et multivariées, et est subdivisée en trois parties distinctes.<p><p>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.<p><p>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.<p><p>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. / Doctorat en Sciences / info:eu-repo/semantics/nonPublished

In this thesis, we present some new results in distribution theory for both discrete and continuous random variables, together with their motivating applications. We start with some results about the Multivariate Gaussian Distribution and its characterization as a maximizer of the Strichartz Estimates. Then, we present some characterizations of discrete and continuous distributions through ideas coming from optimal transportation. After this, we pass to the Simpson's Paradox and see that it is ubiquitous and it appears in Quantum Mechanics as well. We conclude with a group of results about discrete and continuous distributions invariant under symmetries, in particular invariant under the groups $A_1$, an elliptical version of $O(n)$ and $\mathbb{T}^n$. As mentioned, all the results proved in this thesis are motivated by their applications in different research areas. The applications will be thoroughly discussed. We have tried to keep each chapter self-contained and recalled results from other chapters when needed. The following is a more precise summary of the results discussed in each chapter. In chapter \ref{chapter 2}, we discuss a variational characterization of the Multivariate Normal distribution (MVN) as a maximizer of the Strichartz Estimates. Strichartz Estimates appear as a fundamental tool in the proof of wellposedness results for dispersive PDEs. With respect to the characterization of the MVN distribution as a maximizer of the entropy functional, the characterization as a maximizer of the Strichartz Estimate does not require the constraint of fixed variance. In this chapter, we compute the precise optimal constant for the whole range of Strichartz admissible exponents, discuss the connection of this problem to Restriction Theorems in Fourier analysis and give some statistical properties of the family of Gaussian Distributions which maximize the Strichartz estimates, such as Fisher Information, Index of Dispersion and Stochastic Ordering. We conclude this chapter presenting an optimization algorithm to compute numerically the maximizers. Chapter \ref{chapter 3} is devoted to the characterization of distributions by means of techniques from Optimal Transportation and the Monge-Amp\`{e}re equation. We give emphasis to methods to do statistical inference for distributions that do not possess good regularity, decay or integrability properties. For example, distributions which do not admit a finite expected value, such as the Cauchy distribution. The main tool used here is a modified version of the characteristic function (a particular case of the Fourier Transform). An important motivation to develop these tools come from Big Data analysis and in particular the Consensus Monte Carlo Algorithm. In chapter \ref{chapter 4}, we study the \emph{Simpson's Paradox}. The \emph{Simpson's Paradox} is the phenomenon that appears in some datasets, where subgroups with a common trend (say, all negative trend) show the reverse trend when they are aggregated (say, positive trend). Even if this issue has an elementary mathematical explanation, the statistical implications are deep. Basic examples appear in arithmetic, geometry, linear algebra, statistics, game theory, sociology (e.g. gender bias in the graduate school admission process) and so on and so forth. In our new results, we prove the occurrence of the \emph{Simpson's Paradox} in Quantum Mechanics. In particular, we prove that the \emph{Simpson's Paradox} occurs for solutions of the \emph{Quantum Harmonic Oscillator} both in the stationary case and in the non-stationary case. We prove that the phenomenon is not isolated and that it appears (asymptotically) in the context of the \emph{Nonlinear Schr\"{o}dinger Equation} as well. The likelihood of the \emph{Simpson's Paradox} in Quantum Mechanics and the physical implications are also discussed. Chapter \ref{chapter 5} contains some new results about distributions with symmetries. We first discuss a result on symmetric order statistics. We prove that the symmetry of any of the order statistics is equivalent to the symmetry of the underlying distribution. Then, we characterize elliptical distributions through group invariance and give some properties. Finally, we study geometric probability distributions on the torus with applications to molecular biology. In particular, we introduce a new family of distributions generated through stereographic projection, give several properties of them and compare them with the Von-Mises distribution and its multivariate extensions. / Thesis / Doctor of Philosophy (PhD)