Doctor of Philosophy / Department of Statistics / Weixing Song / The regression model has been given a considerable amount of attention and played a significant role in data analysis. The usual assumption in regression analysis is that the variances of the error terms are constant across the data. Occasionally, this assumption of homoscedasticity on the variance is violated; and the data generated from real world applications exhibit heteroscedasticity. The practical importance of detecting heteroscedasticity in regression analysis is widely recognized in many applications because efficient inference for the regression function requires unequal variance to be taken into account. The goal of this thesis is to propose new testing procedures to assess the adequacy of fitting parametric variance function in heteroscedastic regression models. The proposed tests are established in Chapter 2 using certain minimized L[subscript]2 distance between a nonparametric and a parametric variance function estimators. The asymptotic distribution of the test statistics corresponding to the minimum distance estimator under the fixed model and that of the corresponding minimum distance estimators are shown to be normal. These estimators turn out to be [sqrt]n consistent. The asymptotic power of the proposed test against some local nonparametric alternatives is also investigated. Numerical simulation studies are employed to evaluate the nite sample performance of the test in one dimensional and two dimensional cases. The minimum distance method in Chapter 2 requires the calculation of the integrals in the test statistics. These integrals usually do not have a tractable form. Therefore, some numerical integration methods are needed to approximate the integrations. Chapter 3 discusses a nonparametric empirical smoothing lack-of-fit test for the functional form of the variance in regression models that do not involve evaluation of integrals. empirical smoothing lack-of-fit test can be treated as a nontrivial modification of Zheng (1996)'s nonparametric smoothing test and Koul and Ni (2004)'s minimum distance test for the mean function in the classic regression models. The asymptotic normality of the proposed test under the null hypothesis is established. Consistency at some fixed alternatives and asymptotic power under some local alternatives are also discussed. Simulation studies are conducted to assess the nite sample performance of the test. The simulation studies show that the proposed empirical smoothing test is more powerful and computationally more efficient than the minimum distance test and Wang and Zhou (2006)'s test.

Cette thèse est consacrée au problème d'estimation de la fonction de dérive de certains modèles de processus stochastiques périodiques lorsque la durée d'observation tend vers l'infini. Aucune hypothèse de récurrence n'est posée a priori.Dans un premier temps nous considérons le modèle du type signal plus bruit dζt = f (t, θ)dt + σ(t)dWt,; et puis nous étudions l'estimation du paramètre θ à partir d'une observation continue et puis d'une observation discrète du processus {ζt} sur l'intervalle [0; T]. Les fonctions f (·, ·) et σ(·) sont continues et périodiques en t de même période P > 0, σ(·) > 0 et θ ∈ Θ ⊂R. Nous établissons la convergence en probabilité d'un estimateur du maximum de vraisemblance θˆT , sa normalité asymptotique et son efficacité asymptotique minimax. Lorsque f (t, θ) = θf (t), l'expression de θˆT est explicite et nous obtenons la convergence en moyenne quadratique aussi bien pour le cas d'une observation continue que pour le cas d'une observation discrète. De plus, nous déduisons la convergence presque sûre dans le cas d'une observation continue.Dans la seconde partie nous traitons l'estimation non-paramétrique de la fonction f(_) pour les modèles périodiques du type signal plus bruit et du type Ornstein-Uhlenbeck donnés par dζt = f (t)dt + σ(t)dWt, dξt = f (t)ξtdt + dWt. Pour le premier modèle, un estimateur à noyau périodique est construit, la convergence en moyenne quadratique uniformément sur [0; P] et presque sûre de cet estimateur est établie ainsi que sa normalité asymptotique. Dans le cas du modèle d'Ornstein-Uhlenbeck, la convergence du biais ainsi que la convergence en moyenne quadratique uniformément sur [0; P] sont prouvées, et leurs vitesses de convergence sont étudiées. / In this thesis, we consider a drift estimation problem of a certain class of stochastic periodic processes when the length of observation goes to infinity. Firstly, we deal with the linear periodic signal plus noise model dζt = f (t, θ)dt + σ(t)dWt, ;and we study the parametric estimation from a continuous and discrete observation of the process f_tg throughout the interval [0; T]. Using the maximum likelihood method we show the existence of an estimator θˆT which is consistent, asymptotically normal and asymptotically efficient in the sens minimax. When f(t; _) = _f(t), the expression of ^_T is explicit and we obtain the mean square convergence in the both continuous and discrete observation cases. In addition, we deduce the strong consistency in the case of continuous observation.Secondly, we consider the nonparametric estimation problem of the function f(_) for the next two periodic models of type signal plus noise and Ornstein-Uhlenbeckd_t = f(t)dt + _(t)dWt; d_t = f(t)_tdt + dWt:For the signal plus noise model, we build a kernel estimator, the convergence in mean square uniformly over [0; P] and almost sure convergence are established, as well as the asymptotic normality. For the Ornstein-Uhlenbeck model, we prove the convergence uniformly over [0; P] of the bias and the mean square convergence. Moreover, we study the speed of these convergences.

No description available.

We consider a nonparametric survival model with random censoring. To test whether the hazard rate has a parametric form the unknown hazard rate is estimated by a kernel estimator. Based on a limit theorem stating the asymptotic normality of the quadratic distance of this estimator from the smoothed hypothesis an asymptotic ®-test is proposed. Since the test statistic depends on the maximum likelihood estimator for the unknown parameter in the hypothetical model properties of this parameter estimator are investigated. Power considerations complete the approach.

On étudie un système de particules en interactions. Deux types de processus sont utilisés pour modéliser le système. Tout d'abord des processus de Hawkes. On propose deux modèles pour lesquels on obtient l'existence et l'unicité d'une version stationnaire, ainsi qu'une construction graphique de la mesure stationnaire à l'aide d'une décomposition de type Kalikow et d'un algorithme de simulation parfaite.Le deuxième type de processus utilisés est un processus de Markov déterministe par morceaux (PDMP). On montre l'ergodicité de ce processus et propose un estimateur à noyau pour la fonction de taux de saut possédant une vitesse de convergence optimale dans L². / We work on interacting particles systems. Two different types of processes are studied. A first model using Hawkes processes, for which we state existence and uniqueness of a stationnary version. We also propose a graphical construction of the stationnary measure by the mean of a Kalikow-type decomposition and a perfect simulation algorithm.The second model deals with Piecewise deterministic Markov processes (PDMP). We state ergodicity and propose a Kernel estimator for the jump rate function having an optimal speed of convergence in L².

Uma das principais preocupações da macroeconomia é a compreensão da dinâmica da inflação no curto prazo. Entender como a inflação se relaciona com a atividade econômica é decisivo para traçar estratégias de desinflação, assim como, de determinação da trajetória de política monetária. Uma questão que surge é qual a forma exata da relação inflação-produto. Ou seja, podemos caracterizar essa relação como não linear? Se sim, qual a forma dessa não linearidade? Para responder a essas perguntas, estimou-se a relação inflação-produto de forma não-paramétrica através de um local linear kernel estimator. O resultado da estimação gerou uma forma funcional a qual foi aproximada pela estimação, via GMM, de uma curva de Phillips Novo-Keynesiana Híbrida. Essa abordagem foi aplicada para o Brasil a partir de 2000. As estimações sugeriram que a dinâmica da inflação brasileira é melhor descrita quando adiciona-se um termo cúbico relativo ao hiato do produto, ou seja, a inflação brasileira mostrou-se state-dependent. / One of the most important macroeconomics’ concerns is the comprehension about sort-run inflation dynamic. To understand how inflation relates to economic activity is crucial to decision-making in disinflation strategies, as well as in monetary policy paths. A question that arises is what does real form of relation inflation-output trade-off? Could one characterize it as a non-linear relation? If does, what is the shape of this non-linear relation? To answer those questions, we estimate the inflation-output relation non-parametrically using a local linear kernel estimator. The functional form achieved was approximated by a New-Keynesian Hybrid Phillips Curve, which one was estimated by GMM. This approach was applied to Brazil since 2000. We have found evidence that Brazilian inflation dynamic is better described adding a cubic term related to output gap, in other words, the Brazilian inflation is state-dependent.

Uma das principais preocupações da macroeconomia é a compreensão da dinâmica da inflação no curto prazo. Entender como a inflação se relaciona com a atividade econômica é decisivo para traçar estratégias de desinflação, assim como, de determinação da trajetória de política monetária. Uma questão que surge é qual a forma exata da relação inflação-produto. Ou seja, podemos caracterizar essa relação como não linear? Se sim, qual a forma dessa não linearidade? Para responder a essas perguntas, estimou-se a relação inflação-produto de forma não-paramétrica através de um local linear kernel estimator. O resultado da estimação gerou uma forma funcional a qual foi aproximada pela estimação, via GMM, de uma curva de Phillips Novo-Keynesiana Híbrida. Essa abordagem foi aplicada para o Brasil a partir de 2000. As estimações sugeriram que a dinâmica da inflação brasileira é melhor descrita quando adiciona-se um termo cúbico relativo ao hiato do produto, ou seja, a inflação brasileira mostrou-se state-dependent. / One of the most important macroeconomics’ concerns is the comprehension about sort-run inflation dynamic. To understand how inflation relates to economic activity is crucial to decision-making in disinflation strategies, as well as in monetary policy paths. A question that arises is what does real form of relation inflation-output trade-off? Could one characterize it as a non-linear relation? If does, what is the shape of this non-linear relation? To answer those questions, we estimate the inflation-output relation non-parametrically using a local linear kernel estimator. The functional form achieved was approximated by a New-Keynesian Hybrid Phillips Curve, which one was estimated by GMM. This approach was applied to Brazil since 2000. We have found evidence that Brazilian inflation dynamic is better described adding a cubic term related to output gap, in other words, the Brazilian inflation is state-dependent.

Uma das principais preocupações da macroeconomia é a compreensão da dinâmica da inflação no curto prazo. Entender como a inflação se relaciona com a atividade econômica é decisivo para traçar estratégias de desinflação, assim como, de determinação da trajetória de política monetária. Uma questão que surge é qual a forma exata da relação inflação-produto. Ou seja, podemos caracterizar essa relação como não linear? Se sim, qual a forma dessa não linearidade? Para responder a essas perguntas, estimou-se a relação inflação-produto de forma não-paramétrica através de um local linear kernel estimator. O resultado da estimação gerou uma forma funcional a qual foi aproximada pela estimação, via GMM, de uma curva de Phillips Novo-Keynesiana Híbrida. Essa abordagem foi aplicada para o Brasil a partir de 2000. As estimações sugeriram que a dinâmica da inflação brasileira é melhor descrita quando adiciona-se um termo cúbico relativo ao hiato do produto, ou seja, a inflação brasileira mostrou-se state-dependent. / One of the most important macroeconomics’ concerns is the comprehension about sort-run inflation dynamic. To understand how inflation relates to economic activity is crucial to decision-making in disinflation strategies, as well as in monetary policy paths. A question that arises is what does real form of relation inflation-output trade-off? Could one characterize it as a non-linear relation? If does, what is the shape of this non-linear relation? To answer those questions, we estimate the inflation-output relation non-parametrically using a local linear kernel estimator. The functional form achieved was approximated by a New-Keynesian Hybrid Phillips Curve, which one was estimated by GMM. This approach was applied to Brazil since 2000. We have found evidence that Brazilian inflation dynamic is better described adding a cubic term related to output gap, in other words, the Brazilian inflation is state-dependent.

Cette thèse comporte plusieurs procédures d'estimation non-paramétrique de densité de probabilité.Dans chaque cas les variables d'intérêt ne sont pas observées directement, ce qui est une difficulté majeure.La première partie traite un modèle linéaire mixte où des observations répétées sont disponibles.La deuxième partie s'intéresse aux modèles d'équations différentielles stochastiques à effets aléatoires. Plusieurs trajectoires sont observées en temps continu sur un intervalle de temps commun.La troisième partie se place dans un contexte de bruit multiplicatif.Les différentes parties de cette thèse sont reliées par un contexte commun de problème inverse et par une problématique commune: l'estimation de la densité d'une variable cachée. Dans les deux premières parties la densité d'un ou plusieurs effets aléatoires est estimée. Dans la troisième partie il s'agit de reconstruire la densité de la variable d'origine à partir d'observations bruitées.Différentes méthodes d'estimation globale sont utilisées pour construire des estimateurs performants: estimateurs à noyau, estimateurs par projection ou estimateurs construits par déconvolution.La sélection de paramètres mène à des estimateurs adaptatifs et les risques quadratiques intégrés sont majorés grâce à une inégalité de concentration de Talagrand. Une étude sur simulations de chaque estimateur illustre leurs performances. Un jeu de données neuronales est étudié grâce aux procédures mises en place pour les équations différentielles stochastiques. / This thesis contains several nonparametric estimation procedures of a probability density function.In each case, the main difficulty lies in the fact that the variables of interest are not directly observed.The first part deals with a mixed linear model for which repeated observations are available.The second part focuses on stochastic differential equations with random effects. Many trajectories are observed continuously on the same time interval.The third part is in a full multiplicative noise framework.The parts of the thesis are connected by the same context of inverse problems and by a common problematic: the estimation of the density function of a hidden variable.In the first two parts the density of one or two random effects is estimated. In the third part the goal is to rebuild the density of the original variable from the noisy observations.Different global methods are used and lead to well competitive estimators: kernel estimators, projection estimators or estimators built from deconvolution.Parameter selection gives adaptive estimators and the integrated risks are bounded using a Talagrand concentration inequality.A simulation study for each proposed estimator highlights their performances.A neuronal dataset is investigated with the new procedures for stochastic differential equations developed in this work.

Les précipitations et les débits des cours d'eau constituent les deux variables hydrométéorologiques les plus importantes pour l'analyse des bassins versants. Ils fournissent des informations fondamentales pour la gestion intégrée des ressources en eau, telles que l’approvisionnement en eau potable, l'hydroélectricité, les prévisions d'inondations ou de sécheresses ou les systèmes d'irrigation.Dans cette thèse de doctorat sont abordés deux problèmes distincts. Le premier prend sa source dans l’étude des débits des cours d’eau. Dans le but de bien caractériser le comportement global d'un bassin versant, de longues séries temporelles de débit couvrant plusieurs dizaines d'années sont nécessaires. Cependant les données manquantes constatées dans les séries représentent une perte d'information et de fiabilité, et peuvent entraîner une interprétation erronée des caractéristiques statistiques des données. La méthode que nous proposons pour aborder le problème de l'imputation des débits se base sur des modèles de régression dynamique (DRM), plus spécifiquement, une régression linéaire multiple couplée à une modélisation des résidus de type ARIMA. Contrairement aux études antérieures portant sur l'inclusion de variables explicatives multiples ou la modélisation des résidus à partir d'une régression linéaire simple, l'utilisation des DRMs permet de prendre en compte les deux aspects. Nous appliquons cette méthode pour reconstruire les données journalières de débit à huit stations situées dans le bassin versant de la Durance (France), sur une période de 107 ans. En appliquant la méthode proposée, nous parvenons à reconstituer les débits sans utiliser d'autres variables explicatives. Nous comparons les résultats de notre modèle avec ceux obtenus à partir d'un modèle complexe basé sur les analogues et la modélisation hydrologique et d'une approche basée sur le plus proche voisin. Dans la majorité des cas, les DRMs montrent une meilleure performance lors de la reconstitution de périodes de données manquantes de tailles différentes, dans certains cas pouvant allant jusqu'à 20 ans.Le deuxième problème que nous considérons dans cette thèse concerne la modélisation statistique des quantités de précipitations. La recherche dans ce domaine est actuellement très active car la distribution des précipitations exhibe une queue supérieure lourde et, au début de cette thèse, il n'existait aucune méthode satisfaisante permettant de modéliser toute la gamme des précipitations. Récemment, une nouvelle classe de distribution paramétrique, appelée distribution généralisée de Pareto étendue (EGPD), a été développée dans ce but. Cette distribution exhibe une meilleure performance, mais elle manque de flexibilité pour modéliser la partie centrale de la distribution. Dans le but d’améliorer la flexibilité, nous développons, deux nouveaux modèles reposant sur des méthodes semiparamétriques.Le premier estimateur développé transforme d'abord les données avec la distribution cumulative EGPD puis estime la densité des données transformées en appliquant un estimateur nonparamétrique par noyau. Nous comparons les résultats de la méthode proposée avec ceux obtenus en appliquant la distribution EGPD paramétrique sur plusieurs simulations, ainsi que sur deux séries de précipitations au sud-est de la France. Les résultats montrent que la méthode proposée se comporte mieux que l'EGPD, l’erreur absolue moyenne intégrée (MIAE) de la densité étant dans tous les cas presque deux fois inférieure.Le deuxième modèle considère une distribution EGPD semiparamétrique basée sur les polynômes de Bernstein. Plus précisément, nous utilisons un mélange creuse de densités béta. De même, nous comparons nos résultats avec ceux obtenus par la distribution EGPD paramétrique sur des jeux de données simulés et réels. Comme précédemment, le MIAE de la densité est considérablement réduit, cet effet étant encore plus évident à mesure que la taille de l'échantillon augmente. / Precipitation and streamflow are the two most important meteorological and hydrological variables when analyzing river watersheds. They provide fundamental insights for water resources management, design, or planning, such as urban water supplies, hydropower, forecast of flood or droughts events, or irrigation systems for agriculture.In this PhD thesis we approach two different problems. The first one originates from the study of observed streamflow data. In order to properly characterize the overall behavior of a watershed, long datasets spanning tens of years are needed. However, the quality of the measurement dataset decreases the further we go back in time, and blocks of data of different lengths are missing from the dataset. These missing intervals represent a loss of information and can cause erroneous summary data interpretation or unreliable scientific analysis.The method that we propose for approaching the problem of streamflow imputation is based on dynamic regression models (DRMs), more specifically, a multiple linear regression with ARIMA residual modeling. Unlike previous studies that address either the inclusion of multiple explanatory variables or the modeling of the residuals from a simple linear regression, the use of DRMs allows to take into account both aspects. We apply this method for reconstructing the data of eight stations situated in the Durance watershed in the south-east of France, each containing daily streamflow measurements over a period of 107 years. By applying the proposed method, we manage to reconstruct the data without making use of additional variables, like other models require. We compare the results of our model with the ones obtained from a complex approach based on analogs coupled to a hydrological model and a nearest-neighbor approach, respectively. In the majority of cases, DRMs show an increased performance when reconstructing missing values blocks of various lengths, in some of the cases ranging up to 20 years.The second problem that we approach in this PhD thesis addresses the statistical modeling of precipitation amounts. The research area regarding this topic is currently very active as the distribution of precipitation is a heavy-tailed one, and at the moment, there is no general method for modeling the entire range of data with high performance. Recently, in order to propose a method that models the full-range precipitation amounts, a new class of distribution called extended generalized Pareto distribution (EGPD) was introduced, specifically with focus on the EGPD models based on parametric families. These models provide an improved performance when compared to previously proposed distributions, however, they lack flexibility in modeling the bulk of the distribution. We want to improve, through, this aspect by proposing in the second part of the thesis, two new models relying on semiparametric methods.The first method that we develop is the transformed kernel estimator based on the EGPD transformation. That is, we propose an estimator obtained by, first, transforming the data with the EGPD cdf, and then, estimating the density of the transformed data by applying a nonparametric kernel density estimator. We compare the results of the proposed method with the ones obtained by applying EGPD on several simulated scenarios, as well as on two precipitation datasets from south-east of France. The results show that the proposed method behaves better than parametric EGPD, the MIAE of the density being in all the cases almost twice as small.A second approach consists of a new model from the general EGPD class, i.e., we consider a semiparametric EGPD based on Bernstein polynomials, more specifically, we use a sparse mixture of beta densities. Once again, we compare our results with the ones obtained by EGPD on both simulated and real datasets. As before, the MIAE of the density is considerably reduced, this effect being even more obvious as the sample size increases.