Thesis (MComm (Statistics and Actuarial Science))--Stellenbosch University, 2008. / The goodness-of- t of a statistical model describes how well it ts a set of observations. Measures of goodness-of- t typically summarize the discrepancy between observed values and the values expected under the model in question. Such measures can be used in statistical hypothesis testing, for example to test for normality, to test whether two samples are drawn from identical distributions, or whether outcome frequencies follow a speci ed distribution. Goodness-of- t for copulas is a special case of the more general problem of testing multivariate models, but is complicated due to the di culty of specifying marginal distributions. In this thesis, the goodness-of- t test statistics for general distributions and the tests for copulas are investigated, but prior to that an understanding of copulas and their properties is developed. In fact copulas are useful tools for understanding relationships among multivariate variables, and are important tools for describing the dependence structure between random variables. Several univariate, bivariate and multivariate test statistics are investigated, the emphasis being on tests for normality. Among goodness-of- t tests for copulas, tests based on the probability integral transform, Rosenblatt's transformation, as well as some dimension reduction techniques are considered. Bootstrap procedures are also described. Simulation studies are conducted to rst compare the power of rejection of the null hypothesis of the Clayton copula by four di erent test statistics under the alternative of the Gumbel-Hougaard copula, and also to compare the power of rejection of the null hypothesis of the Gumbel-Hougaard copula under the alternative of the Clayton copula. An application of the described techniques is made to a practical data set.

L’objet du travail est d’étudier les prolongements de sous-copules. Un cas important de l’utilisation de tels prolongements est l’estimation non paramétrique d’une copule par le lissage d’une sous-copule (la copule empirique). Lorsque l’estimateur obtenu est une copule, cet estimateur est un prolongement de la souscopule. La thèse présente au chapitre 2 la construction et la convergence uniforme d’un estimateur bona fide d’une copule ou d’une densité de copule. Cet estimateur est un prolongement de type copule empirique basé sur le lissage par le produit tensoriel de fonctions de répartition splines. Le chapitre 3 donne la caractérisation de l’ensemble des prolongements possibles d’une sous-copule. Ce sujet a été traité par le passé; mais les constructions proposées ne s’appliquent pas à la dépendance dans des espaces très généraux. Le chapitre 4 s’attèle à résoudre le problème suivant posé par [Carley, 2002]. Il s’agit de trouver la borne supérieure des prolongements en dimension 3 d’une sous-copule de domaine fini. / The extension of subcopulas is an important domain. One of possible applications is the nonparametric estimation of a copula: it consists of the smoothing of a subcopula (the empirical copula) while preserving the copulas properties. In Chapter 2, we present an extension of the empirical copula based on the tensor product of splines functions. Our estimators are bona fide estimators of the copula. Chapter 3 tackles the problem of finding all possible extensions of a given subcopula. This subject has been treated in the literature but these characterizations do not apply on very general spaces. Chapter 4 deals with the following problem: finding the expression of the upper bound of the extensions of a finite subcopula in dimension 3.

The following dissertation contains two distinct empirical essays which contribute to the overall field of Financial Economics. Chapter 1, entitled “The Determinants of Dynamic Dependence: An Analysis of Commodity Futures and Equity Markets,” examines the determinants of the dynamic equity-commodity return correlations between five commodity futures sub-sectors (energy, foods and fibers, grains and oilseeds, livestock, and precious metals) and a value-weighted equity market index (S&P 500). The study utilizes the traditional DCC model, as well as three time-varying copulas: (i) the normal copula, (ii) the student’s t copula, and (iii) the rotated-gumbel copula as dependence measures. Subsequently, the determinants of these various dependence measures are explored by analyzing several macroeconomic, financial, and speculation variables over different sample periods. Results indicate that the dynamic equity-commodity correlations for the energy, grains and oilseeds, precious metals, and to a lesser extent the foods and fibers, sub-sectors have become increasingly explainable by broad macroeconomic and financial market indicators, particularly after May 2003. Furthermore, these variables exhibit heterogeneous effects in terms of both magnitude and sign on each sub-sectors’ equity-commodity correlation structure. Interestingly, the effects of increased financial market speculation are found to be extremely varied among the five sub-sectors. These results have important implications for portfolio selection, price formation, and risk management. Chapter 2, entitled, “US Community Bank Failure: An Empirical Investigation,” examines the declining, but still pivotal role, of the US community banking industry. The study utilizes survival analysis to determine which accounting and macroeconomic variables help to predict community bank failure. Federal Deposit Insurance Corporation and Federal Reserve Bank data are utilized to compare 452 community banks which failed between 2000 and 2013, relative to a sample of surviving community banks. Empirical results indicate that smaller banks are less likely to fail than their larger community bank counterparts. Additionally, several unique bank-specific indicators of failure emerge which relate to asset quality and liquidity, as well as earnings ratios. Moreover, results show that the use of the macroeconomic indicator of liquidity, the TED spread, provides a substantial improvement in modeling predictive community bank failure.

Ces dernières décennies, nous avons assisté à l'émergence du concept de copule en modélisation statistique. Les copules permettent de faire une analyse séparée des marges et de la structure de dépendance induite par une distribution statistique. Cette séparation facilite l'incorporation de lois non gaussiennes, et en particulier la prise en compte des dépendances non linéaires entre les variables aléatoires. La finance et l'hydrologie sont deux exemples de sciences où les copules sont très utilisées. Cependant, bien qu'il existe beaucoup de familles de copules bivariées, le choix reste limité en plus grande dimension: la construction de copules multivariées/en grande dimension reste un problème ouvert aujourd'hui. Cette thèse présente trois contributions à la modélisation et à l'inférence de copules en grande dimension. Le premier modèle proposé s'écrit comme un produit de copules bivariées, où chaque copule bivariée se combine aux autres via un graphe en arbre. Elle permet de prendre en compte les différents degrés de dépendance entre les différentes paires. La seconde copule est un modèle à facteurs basé sur une classe nonparamétrique de copules bivariées. Elle permet d'obtenir un bon équilibre entre flexibilité et facilité d'utilisation. Cette thèse traite également de l'inférence paramétrique de copules dans le cas général, en établissant les propriétés asymptotiques d'un estimateur des moindres carrés pondérés basé sur les coefficients de dépendance. Les modèles et méthodes proposés sont appliqués sur des données hydrologiques (pluies et débits de rivières). / In the last decades, copulas have been more and more used in statistical modeling. Their popularity owes much to the fact that they allow to separate the analysis of the margins from the analysis of the dependence structure induced by the underlying distribution. This renders easier the modeling of non Gaussian distributions, and, in particular, it allows to take into account non linear dependencies between random variables. Finance and hydrology are two examples of scientific fields where the use of copulas is nowadays standard. However, while many bivariate families exist in the literature, multivariate/high dimensional copulas are much more difficult to construct. This thesis presents three contributions to copula modeling and inference, with an emphasis on high dimensional problems. The first model writes as a product of bivariate copulas and is underlain by a tree structure where each edge represents a bivariate copula. Hence, we are able to model different pairs with different dependence properties. The second one is a factor model built on a nonparametric class of bivariate copulas. It exhibits a good balance between tractability and flexibility. This thesis also deals with the parametric inference of copula models in general. Indeed, the asymptotic properties of a weighted least-squares estimator based on dependence coefficients are established. The models and methods have been applied to hydrological data (flow rates and rain falls).

Motivados por um conjunto de dados de pacientes com insuficiência renal crônica (IRC), propomos uma nova modelagem bayesiana que envolve cópulas da família Arquimediana e um modelo misto para dados de sobrevivência com riscos semicompetitivos. A estrutura de riscos semicompetitivos é bastante comum em estudos clínicos em que dois eventos são de interesse, um intermediário e outro terminal, de forma tal que a ocorrência do evento terminal impede a ocorrência do intermediário mas não vice-versa. Nesta modelagem provamos que a distribuição a posteriori sob a cópula de Clayton é própria. Implementamos os algoritmos de dados aumentados e amostrador de Gibbs para a inferência bayesiana, assim como os criterios de comparação de modelos: LPML, DIC e BIC. Realizamos um estudo de simulação para avaliar o desempenho da modelagem e finalmente aplicamos a metodologia proposta para analisar os dados dos pacientes com IRC, além de outros de pacientes que receberam transplante de medula óssea. / Motivated by a dataset of patients with chronic kidney disease (CKD), we propose a new bayesian model including the Arquimedean copula and a mixed model for survival data with semicompeting risks. The structure of semicompeting risks appears frequently in clinical studies where two-types of events are involved: a nonterminal and a terminal event such that the occurrence of terminal event precludes the occurrence of the non-terminal event but not viceversa. In this work we prove that the posterior distribution is proper when the Clayton copula is used. We implement the data augmentation algorithm and Gibbs sampling for the bayesian inference, as well as some bayesian model selection criteria: LPML, BIC and DIC. We carry out a simulation study for assess the model performance and finally, our methodology is illustrated with the chronic kidney disease study.

Os mercados financeiros são de extrema relevância para as diversas economias do mundo. Sua efetividade na atração de capitais e investimentos é notória. Atualmente, o fluxo financeiro entre os diversos países é muito intenso, devido ao fenômeno da globalização. Tal situação provoca transmissão de crises financeiras entre diferentes países. Neste contexto, a avaliação de contágio financeiro torna-se um tema bastante relevante. A presente dissertação almejou verificar se houve contágio financeiro da crise americana de 2008 sobre os países do BRIC (Brasil, Rússia, Índia e China). Para tanto, foram utilizadas duas metodologias distintas. Uma delas, devido a Fermanian et al. (2002), foi empregada para estimação não paramétrica das cópulas via kernel. Assim, pode-se averiguar se houve aumento significativo nas medidas de dependência. A outra, desenvolvida por Remillard e Scaillet (2009), é um teste de comparação entre duas cópulas empíricas que investiga se houve mudança na estrutura de dependência no período de crise. Os dois procedimentos metodológicos indicaram a ocorrência de contágio da crise americana de 2008 sobre todos os países do BRIC. / Financial markets are extremely relevant to the world's diverse economies. Its effectiveness in attracting capital and investments is notorious. Currently, the financial flow between the various countries is very intense, due to the phenomenon of globalization. This situation leads to the transmission of financial crises between different countries. In this context, the evaluation of financial contagion becomes a very relevant issue. The present dissertation aimed to verify if there was financial contagion of the 2008 US crisis on the BRIC countries (Brazil, Russia, India and China). For that, two different methodologies were used. One of them, due to Fermanian et al. (2002), was used for non-parametric estimation of copula via kernel. Thus, it can be verified if there was a significant increase in the measures of dependence. The other, developed by Remillard and Scaillet (2009), is a test of comparison between two empirical copulas that investigates if there was a change in the dependency structure in the crisis period. The two methodological procedures indicated the occurrence of contagion of the American crisis of 2008 on all BRIC countries.

Este trabalho testa, através da metodologia de cópulas, a hipótese de contágio financeiro entre ações brasileiras e índices de mercado dos países que deram origem às crises do Terror em 2001, da Argentina em 2001, dos Subpprimes em 2007 e do Débito Soberano Europeu em 2009. Além disso, ainda é feita uma análise dos setores econômicos que mais foram afetados por cada crise. Os testes da crise do Terror apresentaram evidências de contágio do SP500 para 24 ações brasileiras, afetando, principalmente os setores ligado à indústria e à energia. As crises da Argentina e do Débito Soberano Europeu apresentaram evidências de contágio dos índices Merval e Athex para apenas 3 empresas. A crise dos Subprimes apresentou evidências de contágio do SP500 para 35 empresas brasileiras, sendo a maioria ligada aos setores financeiros, de energia e industrial. 7 ações foram afetadas pelas duas crises norteamericanas. Os resultados reforçam a importância da análise de contágio em cada empresa individual, ao invés de utilizar o índice do mercado brasileiro como um todo. / This paper tests, through the copulas methodology, the hypothesis of financial contagion between the individual Brazilian stocks and the market indices of the countries where the crises were originated. The crises analyzed are the Terror crisis in 2001, the Argentina’s crisis in 2001, the Subprime crisis in 2007 and the Sovereign Debt crisis in 2009. In addition to this, the Brazilian economic sectors are examined in order to find out which were most affected by each crisis. The tests of the Terror crisis presented evidence of SP500 contagion to 24 Brazilian stocks, affecting, mainly, sectors related to industry and energy. The Argentina’s crisis and the European Sovereign Debt crisis presented contagion’s evidence of the Merval and Athex indices for only 3 Brazilian companies. The Subprimes crisis presented evidence of SP500 contagion for 35 Brazilian companies, mostly related to the financial, energy and industrial sectors. 7 Brazilian stocks were affected by both American crises. The results reinforce the importance of contagion analysis in each individual company, rather than using the Brazilian market index.

Motivados por um conjunto de dados de pacientes com insuficiência renal crônica (IRC), propomos uma nova modelagem bayesiana que envolve cópulas da família Arquimediana e um modelo misto para dados de sobrevivência com riscos semicompetitivos. A estrutura de riscos semicompetitivos é bastante comum em estudos clínicos em que dois eventos são de interesse, um intermediário e outro terminal, de forma tal que a ocorrência do evento terminal impede a ocorrência do intermediário mas não vice-versa. Nesta modelagem provamos que a distribuição a posteriori sob a cópula de Clayton é própria. Implementamos os algoritmos de dados aumentados e amostrador de Gibbs para a inferência bayesiana, assim como os criterios de comparação de modelos: LPML, DIC e BIC. Realizamos um estudo de simulação para avaliar o desempenho da modelagem e finalmente aplicamos a metodologia proposta para analisar os dados dos pacientes com IRC, além de outros de pacientes que receberam transplante de medula óssea. / Motivated by a dataset of patients with chronic kidney disease (CKD), we propose a new bayesian model including the Arquimedean copula and a mixed model for survival data with semicompeting risks. The structure of semicompeting risks appears frequently in clinical studies where two-types of events are involved: a nonterminal and a terminal event such that the occurrence of terminal event precludes the occurrence of the non-terminal event but not viceversa. In this work we prove that the posterior distribution is proper when the Clayton copula is used. We implement the data augmentation algorithm and Gibbs sampling for the bayesian inference, as well as some bayesian model selection criteria: LPML, BIC and DIC. We carry out a simulation study for assess the model performance and finally, our methodology is illustrated with the chronic kidney disease study.

As relações de dependência entre variáveis aleatórias é um dos assuntos mais discutidos em probabilidade e estatística e a forma mais abrangente de estudar essas relações é por meio da distribuição conjunta. Nos últimos anos vem crescendo a utilização de cópulas para representar a estrutura de dependência entre variáveis aleatórias em uma distribuição multivariada. Contudo, ainda existe pouca literatura sobre cópulas quando as distribuições marginais são discretas. No presente trabalho será apresentada uma proposta não-paramétrica de estimação da distribuição conjunta bivariada de variáveis aleatórias discretas utilizando cópulas e polinômios de Bernstein. / The relations of dependence between random variables is one of the most discussed topics in probability and statistics and the best way to study these relationships is through the joint distribution. In the last years has increased the use of copulas to represent the dependence structure among random variables in a multivariate distribution. However, there is still little literature on copulas when the marginal distributions are discrete. In this work we present a non-parametric approach for the estimation of the bivariate joint distribution of discrete random variables using copulas and Bernstein polynomials.

Multidimensional frequency models can be used for modeling number of claims from different branches which are somehow dependent on each other. As in the one-dimensional case Poisson distribution and negative binomial distribution are primarily used for modeling multidimensional claim counts data, only they are extended to higher dimensions. The generalization of multi- dimensional distributions is often done using so-called shock variables, where one random variable is included in all dimensions of a random vector which models claim counts. The more comprehensive approach to modeling dependence uses copulas. Comparison of these models is done on a simulated data of number of claims from two different car insurance guarantees.