Doctor of Philosophy / Department of Statistics / Kun Chen and Weixin Yao / This proposal contains two projects that are related to robust mixture models. In the robust project, we propose a new robust mixture of regression models (Bai et al., 2012). The existing methods for tting mixture regression models assume a normal distribution for error and then estimate the regression param- eters by the maximum likelihood estimate (MLE). In this project, we demonstrate that the MLE, like the least squares estimate, is sensitive to outliers and heavy-tailed error distributions. We propose a robust estimation procedure and an EM-type algorithm to estimate the mixture regression models. Using a Monte Carlo simulation study, we demonstrate that the proposed new estimation method is robust and works much better than the MLE when there are outliers or the error distribution has heavy tails. In addition, the proposed robust method works comparably to the MLE when there are no outliers and the error is normal. In the second project, we propose a new robust mixture of linear mixed-effects models. The traditional mixture model with multiple linear mixed effects, assuming Gaussian distribution for random and error parts, is sensitive to outliers. We will propose a mixture of multiple linear mixed t-distributions to robustify the estimation procedure. An EM algorithm is provided to and the MLE under the assumption of t- distributions for error terms and random mixed effects. Furthermore, we propose to adaptively choose the degrees of freedom for the t-distribution using profile likelihood. In the simulation study, we demonstrate that our proposed model works comparably to the traditional estimation method when there are no outliers and the errors and random mixed effects are normally distributed, but works much better if there are outliers or the distributions of the errors and random mixed effects have heavy tails.

No presente trabalho apresentamos o algoritmo de simulacão perfeita CFTP, proposto em Propp & Wilson (1996). Seguindo o trabalho de Philippe & Robert (2003) implementamos o CFTP gerando amostras da distribuicão normal bivariada truncada no quadrante positivo. O algoritmo proposto e comparado com o amostrador de Gibbs e o método de rejeição. Finalmente, apresentamos sugestões para a implementação do CFTP para gerar amostras da distribuição normal truncada em dimensões maiores que dois e a geração de amostras em conjuntos diferente do quadrante positivo. / This project will display the CFTP perfect simulation algorithm presented at Propp & Wilson (1996). According to Philippe & Robert (2003) will be implemented the CFTP providing samples of the bivariate normal distribution truncated at the positive quadrant. The proposed algorithm is compared to the samples generated by Gibbs Sampler and by the rejection sampling ( or acceptance rejection method or \"accept-reject algorithm\"). Finally, suggestions to the implementation of CFTP in order to produce truncated normal distribution samples at bigger dimensions than two and the provide a diferent set of samples from the positive quadrant.

Multivariate volatility models, such as DCC MGARCH, are estimated under assumption of multivariate normal distribution of random variables, while this assumption have been rejected by empirical evidence. Therefore, the estimated conditional correlation may not explain the whole dependence structure, since under non-normality the linear correlation is only one of the dependency measures. The aim of this thesis is to employ a copula function to the DCC MGARCH model, as copulas are able to link non-normal marginal distributions to create corresponding multivariate joint distribution. The copula-based MGARCH model with uncorrelated dependent errors permits to model conditional cor- relation by DCC-MGARCH and dependence by the copula function, sepa- rately and simultaneously. In other words the model aims to explain addi- tional dependence not captured by traditional DCC MGARCH model due to assumption of normality. In the empirical analysis we apply the model on datasets consisting primarily of stocks of the PX Index and on the pair of S&P500 and NASDAQ100 in order to compare the copula-based MGARCH model to traditional DCC MGARCH in terms of capturing the dependency structure. 1

No presente trabalho apresentamos o algoritmo de simulacão perfeita CFTP, proposto em Propp & Wilson (1996). Seguindo o trabalho de Philippe & Robert (2003) implementamos o CFTP gerando amostras da distribuicão normal bivariada truncada no quadrante positivo. O algoritmo proposto e comparado com o amostrador de Gibbs e o método de rejeição. Finalmente, apresentamos sugestões para a implementação do CFTP para gerar amostras da distribuição normal truncada em dimensões maiores que dois e a geração de amostras em conjuntos diferente do quadrante positivo. / This project will display the CFTP perfect simulation algorithm presented at Propp & Wilson (1996). According to Philippe & Robert (2003) will be implemented the CFTP providing samples of the bivariate normal distribution truncated at the positive quadrant. The proposed algorithm is compared to the samples generated by Gibbs Sampler and by the rejection sampling ( or acceptance rejection method or \"accept-reject algorithm\"). Finally, suggestions to the implementation of CFTP in order to produce truncated normal distribution samples at bigger dimensions than two and the provide a diferent set of samples from the positive quadrant.

The paper is devoted to statistical nonparametric estimation of multivariate distribution density. The influence of data pre-clustering on the estimation accuracy of multimodal density is analysed by means of the Monte-Carlo method.

The paper is devoted to statistical nonparametric estimation of multivariate distribution density. The influence of data pre-clustering on the estimation accuracy of multimodal density is analysed by means of the Monte-Carlo method.

It is well known that the generation of random vectors with non-independent components is difficult. Nevertheless, we propose a new and very simple generation algorithm for multivariate linear densities over point-symmetric domains. Among other applications it can be used to design a simple decomposition-rejection algorithm for multivariate concave distributions. / Series: Research Report Series / Department of Statistics and Mathematics

Le but de cette thèse est l'étude de certains aspects d'un marché financier comportant plusieurs actifs risqués et des options écrites sur ces actifs. Dans un premier essai, nous proposons une expression de la distribution implicite du prix d'un actif sous-jacent en fonction du smile de volatilité associé aux options écrites sur cet actif. L'expression obtenue pour la densité implicite prend la forme d'une densité log-normale plus deux termes d'ajustement. La mise en œuvre de ce résultat est ensuite illustrée à travers deux applications pratiques. Dans le deuxième essai, nous obtenons deux caractérisations de l'absence d'opportunité d'arbitrage en termes de fonctions copules. Chacune de ces caractérisations conduit à une méthode de détection des situations d'arbitrage. La première méthode proposée repose sur une propriété particulière des copules de Bernstein. La seconde méthode est valable dans le cas bivarié et tire profit de résultats sur les bornes de Fréchet-Hoeffding en présence d'information additionnelle sur la dépendance. Les résultats de l'utilisation de ces méthodes sur des données empiriques sont présentés. Enfin, dans le troisième essai, nous proposons une approche pour couvrir avec des options sur spread l'exposition au risque de dépendance d'un portefeuille d'options écrites sur deux actifs. L'approche proposée repose sur l'utilisation de deux modèles paramétriques de dépendance que nous introduisons: les copules Power Frank (PF) et Power Student's t (PST). Le fonctionnement et les résultats de l'approche proposée sont illustrés dans une étude numérique. / This thesis is dedicated to the study of a market with several risky assets and options written on these assets. In a first essay, we express the implied distribution of an underlying asset price as a function of its options implied volatility smile. For the density, the obtained expression has the form of a log-normal density plus two adjustment terms. We then explain how to use these results and develop practical applications. In a first application we value a portfolio of digital options and in another application we fit a parametric distribution. In the second essay, we propose a twofold characterization of the absence of arbitrage opportunity in terms of copula functions. We then propose two detection methods. The first method relies on a particular property of Bernstein copulas. The second method, valid only in the case of a market with two risky assets, is based upon results on improved Fréchet-Hoeffding bounds in presence of additional information about the dependence. We also present results obtained with the proposed methods applied to empirical data. Finally, in the third essay, we develop an approach to hedge, with spread options, an exposure to dependence risk for a portfolio comprising two-asset options. The approach we propose is based on two parametric models of dependence that we introduce. These dependence models are copulas functions named Power Frank (PF) and Power Student's t (PST). The results obtained with the proposed approach are detailed in a numerical study.

In this paper, we prove that the joint distribution of random vectors Z 1 and Z 2 and the distribution of Z 2 are skew normal provided that Z 1 is skew normally distributed and Z 2 conditioning on Z 1 is distributed as closed skew normal. Also, we extend the main results to the matrix variate case.