In a nonparametric setting, the functional form of the relationship between the response variable and the associated predictor variables is unspecified; however it is assumed to be a smooth function. The main aim of nonparametric regression is to highlight an important structure in data without any assumptions about the shape of an underlying regression function. In regression, the random and fixed design models should be distinguished. Among the variety of nonparametric regression estimators currently in use, kernel type estimators are most popular. Kernel type estimators provide a flexible class of nonparametric procedures by estimating unknown function as a weighted average using a kernel function. The bandwidth which determines the influence of the kernel has to be adapted to any kernel type estimator. Our focus is on Nadaraya-Watson estimator and Local Linear estimator which belong to a class of kernel type regression estimators called local polynomial kerne l estimators. A closely related problem is the determination of an appropriate sample size that would be required to achieve a desired confidence level of accuracy for the nonparametric regression estimators. Since sequential procedures allow an experimenter to make decisions based on the smallest number of observations without compromising accuracy, application of sequential procedures to a nonparametric regression model at a given point or series of points is considered. The motivation for using such procedures is: in many applications the quality of estimating an underlying regression function in a controlled experiment is paramount; thus, it is reasonable to invoke a sequential procedure of estimation that chooses a sample size based on recorded observations that guarantees a preassigned accuracy. We have employed sequential techniques to develop a procedure for constructing a fixed-width confidence interval for the predicted value at a specific point of the independent variable. These fixed-width confidence intervals are developed using asymptotic properties of both Nadaraya-Watson and local linear kernel estimators of nonparametric kernel regression with data-driven bandwidths and studied for both fixed and random design contexts. The sample sizes for a preset confidence coefficient are optimized using sequential procedures, namely two-stage procedure, modified two-stage procedure and purely sequential procedure. The proposed methodology is first tested by employing a large-scale simulation study. The performance of each kernel estimation method is assessed by comparing their coverage accuracy with corresponding preset confidence coefficients, proximity of computed sample sizes match up to optimal sample sizes and contrasting the estimated values obtained from the two nonparametric methods with act ual values at given series of design points of interest. We also employed the symmetric bootstrap method which is considered as an alternative method of estimating properties of unknown distributions. Resampling is done from a suitably estimated residual distribution and utilizes the percentiles of the approximate distribution to construct confidence intervals for the curve at a set of given design points. A methodology is developed for determining whether it is advantageous to use the symmetric bootstrap method to reduce the extent of oversampling that is normally known to plague Stein's two-stage sequential procedure. The procedure developed is validated using an extensive simulation study and we also explore the asymptotic properties of the relevant estimators. Finally, application of our proposed sequential nonparametric kernel regression methods are made to some problems in software reliability and finance.

O aumento dos fluxos globais comerciais e financeiros, a partir da década de 90, e as diversas crises ocorridas até o atual período fizeram da avaliação de contágio um tema extremamente relevante, tanto para investidores quanto para formuladores de política. Nesse sentido, a presente dissertação tem como objetivo testar a hipótese de contágio financeiro para os mercados de Brasil, Inglaterra e Espanha em face à última crise americana de 2008. Para tanto, desenvolveu-se o artigo que integra o Capítulo 2 - a espinha dorsal deste trabalho - com dados diários dos retornos dos índices de Jan/2004 a Jun/2011. No âmbito da metodologia de cópulas, adotou-se uma estratégia empírica com base em duas etapas: i) a estimativa não paramétrica de cópulas, via kernel, utilizando o método desenvolvido em Fermanian et al. (2002) e a avaliação através de uma abordagem de bootstrap, sobre a ocorrência de um aumento significativo nas medidas de dependência delas extraídas; ii) testes sobre a igualdade entre cópulas empíricas, conforme proposto por Remillard e Scaillet (2009), a fim de verificar se houve mudança na estrutura de dependência a partir da crise. Os resultados obtidos nas duas etapas da estratégia empírica são semelhantes e sugerem a existência de contágio financeiro para os países analisados no período estudado. / The increase in global trade and financial flows since the 90’s, and the various crises in the current period until these days made contagion an extremely important issue for both investors and policy makers. Accordingly, this dissertation aims to test the hypothesis of financial contagion between USA and markets in Brazil, England and Spain in the face of the last USA crisis of 2008. To this end, we produce the article in Chapter 2 - the backbone of this work - with daily data of index-returns from Jan/2004 to Jun/2011. Under the scope of copula methodology, we addopt an empirical strategy based on two steps: i) estimating nonparametric copulas via kernel, using the method developed in Fermanian et al. (2002) and assessing through a bootstrap approach whether a significant change in dependence measures extracts thereof, ii) testing whether two empirical estimated copulas are the same, as proposed by Remillard e Scaillet (2009), to check again whether dependence structures change with crisis. The results obtained in these two steps of the empirical strategy are similar and suggest the existence of financial contagion between the countries analysed in the studied period.

O aumento dos fluxos globais comerciais e financeiros, a partir da década de 90, e as diversas crises ocorridas até o atual período fizeram da avaliação de contágio um tema extremamente relevante, tanto para investidores quanto para formuladores de política. Nesse sentido, a presente dissertação tem como objetivo testar a hipótese de contágio financeiro para os mercados de Brasil, Inglaterra e Espanha em face à última crise americana de 2008. Para tanto, desenvolveu-se o artigo que integra o Capítulo 2 - a espinha dorsal deste trabalho - com dados diários dos retornos dos índices de Jan/2004 a Jun/2011. No âmbito da metodologia de cópulas, adotou-se uma estratégia empírica com base em duas etapas: i) a estimativa não paramétrica de cópulas, via kernel, utilizando o método desenvolvido em Fermanian et al. (2002) e a avaliação através de uma abordagem de bootstrap, sobre a ocorrência de um aumento significativo nas medidas de dependência delas extraídas; ii) testes sobre a igualdade entre cópulas empíricas, conforme proposto por Remillard e Scaillet (2009), a fim de verificar se houve mudança na estrutura de dependência a partir da crise. Os resultados obtidos nas duas etapas da estratégia empírica são semelhantes e sugerem a existência de contágio financeiro para os países analisados no período estudado. / The increase in global trade and financial flows since the 90’s, and the various crises in the current period until these days made contagion an extremely important issue for both investors and policy makers. Accordingly, this dissertation aims to test the hypothesis of financial contagion between USA and markets in Brazil, England and Spain in the face of the last USA crisis of 2008. To this end, we produce the article in Chapter 2 - the backbone of this work - with daily data of index-returns from Jan/2004 to Jun/2011. Under the scope of copula methodology, we addopt an empirical strategy based on two steps: i) estimating nonparametric copulas via kernel, using the method developed in Fermanian et al. (2002) and assessing through a bootstrap approach whether a significant change in dependence measures extracts thereof, ii) testing whether two empirical estimated copulas are the same, as proposed by Remillard e Scaillet (2009), to check again whether dependence structures change with crisis. The results obtained in these two steps of the empirical strategy are similar and suggest the existence of financial contagion between the countries analysed in the studied period.

O aumento dos fluxos globais comerciais e financeiros, a partir da década de 90, e as diversas crises ocorridas até o atual período fizeram da avaliação de contágio um tema extremamente relevante, tanto para investidores quanto para formuladores de política. Nesse sentido, a presente dissertação tem como objetivo testar a hipótese de contágio financeiro para os mercados de Brasil, Inglaterra e Espanha em face à última crise americana de 2008. Para tanto, desenvolveu-se o artigo que integra o Capítulo 2 - a espinha dorsal deste trabalho - com dados diários dos retornos dos índices de Jan/2004 a Jun/2011. No âmbito da metodologia de cópulas, adotou-se uma estratégia empírica com base em duas etapas: i) a estimativa não paramétrica de cópulas, via kernel, utilizando o método desenvolvido em Fermanian et al. (2002) e a avaliação através de uma abordagem de bootstrap, sobre a ocorrência de um aumento significativo nas medidas de dependência delas extraídas; ii) testes sobre a igualdade entre cópulas empíricas, conforme proposto por Remillard e Scaillet (2009), a fim de verificar se houve mudança na estrutura de dependência a partir da crise. Os resultados obtidos nas duas etapas da estratégia empírica são semelhantes e sugerem a existência de contágio financeiro para os países analisados no período estudado. / The increase in global trade and financial flows since the 90’s, and the various crises in the current period until these days made contagion an extremely important issue for both investors and policy makers. Accordingly, this dissertation aims to test the hypothesis of financial contagion between USA and markets in Brazil, England and Spain in the face of the last USA crisis of 2008. To this end, we produce the article in Chapter 2 - the backbone of this work - with daily data of index-returns from Jan/2004 to Jun/2011. Under the scope of copula methodology, we addopt an empirical strategy based on two steps: i) estimating nonparametric copulas via kernel, using the method developed in Fermanian et al. (2002) and assessing through a bootstrap approach whether a significant change in dependence measures extracts thereof, ii) testing whether two empirical estimated copulas are the same, as proposed by Remillard e Scaillet (2009), to check again whether dependence structures change with crisis. The results obtained in these two steps of the empirical strategy are similar and suggest the existence of financial contagion between the countries analysed in the studied period.

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.

As modern applications of machine learning and data mining are forced to deal with ever more massive quantities of data, practitioners quickly run into difficulty with the scalability of even the most basic and fundamental methods. We propose to provide scalability through a marriage between classical, empirical-style Monte Carlo approximation and deterministic multi-tree techniques. This union entails a critical compromise: losing determinism in order to gain speed. In the face of large-scale data, such a compromise is arguably often not only the right but the only choice. We refer to this new approximation methodology as Multi-Tree Monte Carlo. In particular, we have developed the following fast approximation methods: 1. Fast training for kernel conditional density estimation, showing speedups as high as 10⁵ on up to 1 million points. 2. Fast training for general kernel estimators (kernel density estimation, kernel regression, etc.), showing speedups as high as 10⁶ on tens of millions of points. 3. Fast singular value decomposition, showing speedups as high as 10⁵ on matrices containing billions of entries. The level of acceleration we have shown represents improvement over the prior state of the art by several orders of magnitude. Such improvement entails a qualitative shift, a commoditization, that opens doors to new applications and methods that were previously invisible, outside the realm of practicality. Further, we show how these particular approximation methods can be unified in a Multi-Tree Monte Carlo meta-algorithm which lends itself as scaffolding to the further development of new fast approximation methods. Thus, our contribution includes not just the particular algorithms we have derived but also the Multi-Tree Monte Carlo methodological framework, which we hope will lead to many more fast algorithms that can provide the kind of scalability we have shown here to other important methods from machine learning and related fields.

Nous nous intéressons dans cette thèse aux méthodes d’estimation non paramétriques par noyaux récursifs ainsi qu’à leurs applications à la prévision. Nous introduisons dans un premier chapitre une famille d’estimateurs récursifs de la densité indexée par un paramètre ℓ ∈ [0, 1]. Leur comportement asymptotique en fonction de ℓ va nous amener à introduire des critères de comparaison basés sur les biais, variance et erreur quadratique asymptotiques. Pour ces critères, nous comparons les estimateurs entre eux et aussi comparons notre famille à l’estimateur non récursif de la densité de Parzen-Rosenblatt. Ensuite, nous définissons à partir de notre famille d’estimateurs de la densité, une famille d’estimateurs récursifs à noyau de la fonction de régression. Nous étudions ses propriétés asymptotiques en fonction du paramètre ℓ. Nous utilisons enfin les résultats obtenus sur l’estimation de la régression pour construire un prédicteur non paramétrique par noyau. Nous obtenons ainsi une famille de prédicteurs non paramétriques qui permettent de réduire considérablement le temps de calcul. Des exemples d’application sont donnés pour valider la performance de nos estimateurs / The aim of this thesis is to study methods of nonparametric estimation based on recursive kernel and their applications to forecasting. We introduce in the first chapter a family of recursive density estimators indexed by a parameter ℓ ∈ [0, 1]. We study their asymptotic behavior according to ℓ, and then we introduce criteria of comparison based on bias, variance and asymptotic quadratic error. For these criteria, we compare our estimators in terms of ℓ, and also compare our family to the non-recursive density estimator of Parzen-Rosenblatt. As for density, we define a family of recursive kernel estimators of regression function. We study its asymptotic properties according to the parameter ℓ. Finally, results of regression estimation are applied to define a family of nonparametric predictors that reduce considerably the computing time and examples of application are given to validate the performance of our methods

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.

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.

Cette thèse est consacrée à l'étude de certains modèles de déformation semi-paramétriques. Notre objectif est de proposer des méthodes récursives, issues d'algorithmes stochastiques, pour estimer les paramètres de ces modèles. Dans la première partie, on présente les outils théoriques existants qui nous seront utiles dans la deuxième partie. Dans un premier temps, on présente un panorama général sur les méthodes d'approximation stochastique, en se focalisant en particulier sur les algorithmes de Robbins-Monro et de Kiefer-Wolfowitz. Dans un second temps, on présente les méthodes à noyaux pour l'estimation de fonction de densité ou de régression. On s'intéresse plus particulièrement aux deux estimateurs à noyaux les plus courants qui sont l'estimateur de Parzen-Rosenblatt et l'estimateur de Nadaraya-Watson, en présentant les versions récursives de ces deux estimateurs.Dans la seconde partie, on présente tout d'abord une procédure d'estimation récursive semi-paramétrique du paramètre de translation et de la fonction de régression pour le modèle de translation dans la situation où la fonction de lien est périodique. On généralise ensuite ces techniques au modèle vectoriel de déformation à forme commune en estimant les paramètres de moyenne, de translation et d'échelle, ainsi que la fonction de régression. On s'intéresse finalement au modèle de déformation paramétrique de variables aléatoires dans le cadre où la déformation est connue à un paramètre réel près. Pour ces trois modèles, on établit la convergence presque sûre ainsi que la normalité asymptotique des estimateurs paramétriques et non paramétriques proposés. Enfin, on illustre numériquement le comportement de nos estimateurs sur des données simulées et des données réelles. / This thesis is devoted to the study of some semi-parametric deformation models.Our aim is to provide recursive methods, related to stochastic algorithms, in order to estimate the different parameters of the models. In the first part, we present the theoretical tools which we will use in the next part. On the one hand, we focus on stochastic approximation methods, in particular the Robbins-Monro algorithm and the Kiefer-Wolfowitz algorithm. On the other hand, we introduce kernel estimators in order to estimate a probability density function and a regression function. More particularly, we present the two most famous kernel estimators which are the one of Parzen-Rosenblatt and the one of Nadaraya-Watson. We also present their recursive version.In the second part, we present the results we obtained in this thesis.Firstly, we provide a recursive estimation method of the shift parameter and the regression function for the translation model in which the regression function is periodic. Secondly, we extend this estimation procedure to the shape invariant model, providing estimation of the height parameter, the translation parameter and the scale parameter, as well as the common shape function.Thirdly, we are interested in the parametric deformation model of random variables where the deformation is known and depending on an unknown parameter.For these three models, we establish the almost sure convergence and the asymptotic normality of each estimator. Finally, we numerically illustrate the asymptotic behaviour of our estimators on simulated data and on real data.