In this thesis, we consider several statistical techniques applicable to claim frequency models of an MTPL portfolio with a focus on overdispersion. The practical part of the work is focused on the application and comparison of the models on real data represented by an MTPL portfolio. The comparison is presented by the results of goodness-of-fit measures. Furthermore, the predictive power of selected models is tested for the given dataset, using the simulation method. Hence, this thesis provides a combination of the analysis of goodness-of-fit results and the predictive power of the models.

Cette thèse porte sur l'estimation de fonctions à l'aide de tests dans trois cadres statistiques différents. Nous commençons par étudier le problème de l'estimation des intensités de processus de Poisson avec covariables. Nous démontrons un théorème général de sélection de modèles et en déduisons des bornes de risque non-asymptotiques sous des hypothèses variées sur la fonction à estimer. Nous estimons ensuite la densité de transition d'une chaîne de Markov homogène et proposons pour cela deux procédures. La première, basée sur la sélection d'estimateurs constants par morceaux, permet d'établir une inégalité de type oracle sous des hypothèses minimales sur la chaîne de Markov. Nous en déduisons des vitesses de convergence uniformes sur des boules d'espaces de Besov inhomogènes et montrons que l'estimateur est adaptatif par rapport à la régularité de la densité de transition. La performance de l'estimateur est aussi évalué en pratique grâce à des simulations numériques. La seconde procédure peut difficilement être implémenté en pratique mais permet d'obtenir un résultat général de sélection de modèles et d'en déduire des vitesses de convergence sous des hypothèses plus générales sur la densité de transition. Finalement, nous proposons un nouvel estimateur paramétrique d'une densité. Son risque est contrôlé sous des hypothèses pour lesquelles la méthode du maximum de vraisemblance peut ne pas fonctionner. Les simulations montrent que ces deux estimateurs sont très proches lorsque le modèle est vrai et suffisamment régulier. Il est cependant robuste, contrairement à l'estimateur du maximum de vraisemblance. / This thesis deals with the estimation of functions from tests in three statistical settings. We begin by studying the problem of estimating the intensities of Poisson processes with covariates. We prove a general model selection theorem from which we derive non-asymptotic risk bounds under various assumptions on the target function. We then propose two procedures to estimate the transition density of an homogeneous Markov chain. The first one selects an estimator among a collection of piecewise constant estimators. The selected estimator is shown to satisfy an oracle-type inequality under minimal assumptions on the Markov chain which allows us to deduce uniform rates of convergence over balls of inhomogeneous Besov spaces. Besides, the estimator is adaptive with respect to the smoothness of the transition density. We also evaluate the performance of the estimator in practice by carrying out numerical simulations. The second procedure is only of theoretical interest but yields a general model selection theorem from which we derive rates of convergence under more general assumptions on the transition density. Finally, we propose a new parametric estimator of a density. We upper-bound its risk under assumptions for which the maximum likelihood method may not work. The simulations show that these two estimators are very close when the model is true and regular enough. However, contrary to the maximum likelihood estimator, this estimator is robust.

No description available.

Nesta dissertação estudou-se o modelo GARMA para modelar séries temporais de dados de contagem com as distribuições condicionais de Poisson, binomial e binomial negativa. A principal finalidade foi analisar no contexto clássico e bayesiano, o desempenho e a qualidade do ajuste dos modelos de interesse, bem como o desempenho dos percentis de cobertura dos intervalos de confiança dos parâmetros para os modelos adotados. Para atingir tal finalidade considerou-se a análise dos estimadores pontuais bayesianos e foram analisados intervalos de credibilidade. Neste estudo é proposta uma distribuição a priori conjugada para os parâmetros dos modelos e busca-se a distribuição a posteriori, a qual associada a certas funções de perda permite encontrar estimativas bayesianas para os parâmetros. Na abordagem clássica foram calculados estimadores de máxima verossimilhança, usandose o método de score de Fisher e verificou-se por meio de simulação a consistência dos mesmos. Com os estudos desenvolvidos pode-se observar que, tanto a inferência clássica quanto a inferência bayesiana para os parâmetros dos modelos em questão, apresentou boas propriedades analisadas por meio das propriedades dos estimadores pontuais. A última etapa do trabalho consiste na análise de um conjunto de dados reais, sendo uma série real correspondente ao número de internações por causa da dengue em Campina Grande. Estes resultados mostram que tanto o estudo clássico, quanto o bayesiano, são capazes de descrever bem o comportamento da série / In this work, it was studied the GARMA model to model time series count data with Poisson, binomial and negative binomial discrete conditional distributions. The main goal is to analyze, in the bayesian and classic context, the performance and the quality of fit of the corresponding models, as well as the coverage percentages performance to these models. To achieve this purpose we considered the analysis of Bayesian estimators and credible intervals were analyzed. To the Bayesian study it was proposed a priori distribution joined to the models parameters and sought a posteriori distribution, which one associate with to certain loss functions allows finding out Bayesian estimates to the parameters. In the classical approach, it was calculated the maximum likelihood estimators using the method of Fisher scoring, whose interest was to verify, by simulation, the consistence. With the studies developed we can notice that, both classical and inference Bayesian inference for the parameters of those models, presented good properties analysed through the properties of the punctual estimators. The last stage of the work consisted of the analysis of one real data set, being a real serie corresponding to the admission number because of dengue in the city of Campina Grande. These results show that both the classic and the Bayesian studies are able to describe well the behavior of the serie

Membranas e organelas constituem estruturas presentes nas células dos organismos. Estas estruturas representam interfaces entre eletrólitos. Uma tentativa de descrever, interpretar e compreender a distribuição iônica nas vizinhanças destas estruturas é feita neste trabalho com a análise de resultados de experimentos obtidos na investigação de alguns sistemas: 1) Sistemas constituídos por micelas de SDS. Medidas do pH nas vizinhanças de superfícies das micelas por sondas derivadas do ácido salicílico mostra variações do pH em termos da concentração de SDS e da concentração de sal adicionado. O objetivo dos experimentos é inferir o comportamento do pH nas vizinhanças de membranas biológicas que, por dissociação de alguns fosfolipídios, podem apresentar segmentos da membrana com carga na superfície. 2) A natureza \"zwitteriônica\" das membranas biológicas motivou o estudo da \"ligação iônica\" em micelas \"zwitteriônicas\", imersas em soluções com eletrólito, em concentrações variadas de sais de Cl- e Br- com os cátions monovalentes, Li+, Na+, K+, Rb+, Cs+ e tetrametil amônio (TMA+) e bivalentes Mg2+ e Ca2+. Os experimentos consistiram em determinar a concentração de haletos próximos a micela. A técnica de captura química mostra que há um grau de seletividade que não é determinado apenas pela carga iônica. 3) Resultados preliminares do grau de dissociação no interior de vesiculas de, cloreto de dimetildioctadecil amônio, DODAC, indicam que 8% dos monômeros estão dissociados em vesículas com o diâmetro médio em torno de 150 e 300 nm. As leis fundamentais usadas para compreender os resultados estão aliadas a hipótese de que os sistemas estudados estão em equilibrio termodinâmico, que a interação eletrostática é predominante e ao potencial eletrostático é conferido o papel do potencial da força média que atua nos íons. A simplificação adicional consistindo em admitir que os íons se comportam como cargas puntiformes no que se refere à interação eletrostática conduz o modelo teórico à equação de Poisson-Boltzmann que, linearizada, resulta na equação de Debye-Hückel. Hipóteses adicionais se fazem necessárias para formular o modelo como um problema matemático com condições de contorno. Para cada situação as hipóteses adicionais são discutidas. Sistemas hipotéticos são analizados com o intuito de comparar os resultados provenientes da equação de Poisson-Boltzmann e da equação de Debye-Hückel. A análise teórica dos sistemas conduz a resultados em acordo com os valores medidos. Entre as conclusões obtidas, neste trabalho, são mencionados: 1- As sondas derivadas do ácido salicílico mantêm os grupos dissociáveis próximos a superfície, a distância é da ordem de 0,1 nm, mesmo para sondas que apresentam cadeia longa entre o grupo nitrogênio e o grupo dissociável. 2- A especificidade iônica é bem descrita utilizando, além da carga elétrica, a massa do íon. Os íons na superfície de uma micela zwitteriônica têm liberdade translacional e portanto superfícies zwitteriônicas em solução de eletrólito apresentam condutividade elétrica na superfície. 3- As concentrações iônicas no interior de vesículas são uniformes em praticamente toda a região interna, apresentando variações apenas nas vizinhanças da superfície interna carregada eletricamente. / Membranes and organelles are structures present in biological systems. Such structures are interfaces between electrolytes. Addressing to the description, interpretation and comprehension of the ionic distribution around the structures an attempt is done is this work analyzing experimental results from the investigation of the following systems: 1) SDS micelles. \"pH\" measurements in the micellar surface neighborhood using salicylic acid probes show the pH values dependence with the SDS and added salt concentrations. The experiments aimed to infer the pH behavior in biological membranes where same phospholipids may dissociate and portions of the surface can acquire electrical charges. 2) The zwitterionic nature of biological membranes leads to the investigation of ion binding in zwitterionic micelles in electrolyte solutions with varied concentrations of Cl- or Br<SUP- salts with the cations Li+, Na+, K+, Rb+, Cs+ e tetramethylammonium (TMA+) and the bivalent cations Mg2+ e Ca2+. Halide concentration in the micellar vicinity was measured. Chemical trapping method shows there is a selectivity degree that does not depend only on the ionic charge. 3) Preliminary results in the determination of the inner dissociation degree of dioctadecyldimethylammonium chloride vesicles, DODAC, show that about 8% of the vesicle constituent monomers are dissociated in vesicles with 150 and 300 nm mean diameter. The theoretical description is based upon the thermodynamics equilibrium hypothesis about the systems and that the electrostatic interaction is the stronger interaction and also it is attributed to the mean electrostatic potential the role of the mean force potential acting on the ionic species. A further simplification in considering the ions as point charges with respect to the electrostatic interactions leads the model to the Poisson-Boltzmann equation and under linearization results in the Debye-Hückel alternative description. Additional hypothesis are necessary in order to have the model as a mathematical problem with boundary conditions and are discussed for each system. Hypothetical systems are analysed aiming the comparison between Poisson-Boltzmann and Debye- Hückel descriptions. Some conclusions derived in the analysis are: 1- The salicylic acid probes have the dissociable groups always near (~0,1nm) the micellar surface, even to the probes that have a long chain between the nitrogen group and the dissociable group. 2- In a zwitterionic micelle the ions on the surface have translational freedom and this is way the zwitterionic membranes in electrolyte solutions are conducting surfaces. 3- The ionic concentrations in the vesicle interior have uniformly value almost everywhere showing variations only in the vicinity of the electrically charged interior surface. The theoretical study of the three systems considered gives results in accord with experimental data.

Dados longitudinais na forma de contagens e na forma binária são muito comuns, os quais, frequentemente, podem ser analisados por distribuições de Poisson e de Bernoulli, respectivamente, pertencentes à família exponencial. Duas das principais limitações para modelar esse tipo de dados são: (1) a ocorrência de superdispersão, ou seja, quando a variabilidade dos dados não é adequadamente descrita pelos modelos, que muitas vezes apresentam uma relação pré-estabelecida entre a média e a variância, e (2) a correlação existente entre medidas realizadas repetidas vezes na mesma unidade experimental. Uma forma de acomodar a superdispersão é pela utilização das distribuições binomial negativa e beta binomial, ou seja, pela inclusão de um efeito aleatório com distribuição gama quando se considera dados provenientes de contagens e um efeito aleatório com distribuição beta quando se considera dados binários, ambos introduzidos de forma multiplicativa. Para acomodar a correlação entre as medidas realizadas no mesmo indivíduo podem-se incluir efeitos aleat órios com distribuição normal no preditor linear. Esses situações podem ocorrer separada ou simultaneamente. Molenberghs et al. (2010) propuseram modelos que generalizam os modelos lineares generalizados mistos Poisson-normal e Bernoulli-normal, incorporando aos mesmos a superdispersão. Esses modelos foram formulados e ajustados aos dados, usando-se o método da máxima verossimilhança. Entretanto, para um modelo de efeitos aleatórios, é natural pensar em uma abordagem Bayesiana. Neste trabalho, são apresentados modelos Bayesianos hierárquicos para dados longitudinais, na forma de contagens e binários que apresentam superdispersão. A análise Bayesiana hierárquica é baseada no método de Monte Carlo com Cadeias de Markov (MCMC) e para implementação computacional utilizou-se o software WinBUGS. A metodologia para dados na forma de contagens é usada para a análise de dados de um ensaio clínico em pacientes epilépticos e a metodologia para dados binários é usada para a análise de dados de um ensaio clínico para tratamento de dermatite. / Longitudinal count and binary data are very common, which often can be analyzed by Poisson and Bernoulli distributions, respectively, members of the exponential family. Two of the main limitations to model this data are: (1) the occurrence of overdispersion, i.e., the phenomenon whereby variability in the data is not adequately captured by the model, and (2) the accommodation of data hierarchies owing to, for example, repeatedly measuring the outcome on the same subject. One way of accommodating overdispersion is by using the negative-binomial and beta-binomial distributions, in other words, by the inclusion of a random, gamma-distributed eect when considering count data and a random, beta-distributed eect when considering binary data, both introduced by multiplication. To accommodate the correlation between measurements made in the same individual one can include normal random eects in the linear predictor. These situations can occur separately or simultaneously. Molenberghs et al. (2010) proposed models that simultaneously generalizes the generalized linear mixed models Poisson-normal and Bernoulli-normal, incorporating the overdispersion. These models were formulated and tted to the data using maximum likelihood estimation. However, these models lend themselves naturally to a Bayesian approach as well. In this paper, we present Bayesian hierarchical models for longitudinal count and binary data in the presence of overdispersion. A hierarchical Bayesian analysis is based in the Monte Carlo Markov Chain methods (MCMC) and the software WinBUGS is used for the computational implementation. The methodology for count data is used to analyse a dataset from a clinical trial in epileptic patients and the methodology for binary data is used to analyse a dataset from a clinical trial in toenail infection named onychomycosis.

Une grande part des muqueuses à l’intérieur du corps humain sont recouvertes de cils qui, par leurs mouvements coordonnés, conduisent à une circulation de la couche de fluide nappant la muqueuse. Dans le cas de la paroi interne des bronches, ce processus permet l’évacuation des impuretés inspirées à l’extérieur de l’appareil respiratoire.Dans cette thèse, nous nous intéressons aux effets du ou des cils sur le fluide, en nous plaçant à l’échelle du cil, et on considère pour cela les équations de Stokes incompressible. Due à la finesse du cil, une simulation directe demanderait un raffinement important du maillage au voisinage du cil, pour un maillage qui évoluerait à chaque pas de temps. Cette approche étant trop onéreuse en terme de coûts de calculs, nous avons considéré l’asymptotique d’un diamètre du cil tendant vers 0 et d’une vitesse qui tend vers l’infini : le cil est modélisé par un Dirac linéique de forces en terme source. Nous avons montré qu’il était possible de remplacer ce Dirac linéique par une somme de Dirac ponctuels distribués le long du cil. Ainsi, nous nous sommes ramenés, par linéarité, à étudier le problème de Stokes avec en terme source une force ponctuelle. Si les calculs sont ainsi simplifiés (et leurs coûts réduits), le problème final est lui plus singulier, ce qui motive une analyse numérique fine et l’élaboration d’une nouvelle méthode de résolution.Nous avons d’abord étudié une version scalaire de ce problème : le problème de Poisson avec une masse de Dirac en second membre. La solution exacte étant singulière, la solution éléments finis est à définir avec précaution. La convergence de la méthode étant dégradée dans ce cas-là, par rapport à celle dans le cas régulier, nous nous sommes intéressés à des estimations locales. Nous avons démontré une convergence quasi-optimale en norme Hs (s ě 1) sur un sous-domaine qui exclut la singularité. Des résultats analogues ont été obtenus dans le cas du problème de Stokes.Pour palier les problèmes liés à une mauvais convergence sur l’ensemble du domaine, nous avons élaboré une méthode pour résoudre des problème elliptiques avec une masse de Dirac ou une force ponctuelle en terme source. Basée sur celle des éléments finis standard, elle s’appuie sur la connaissance explicite de la singularité de la solution exacte. Une fois données la position de chacun des cils et leur paramétrisation, notre méthode rend possible la simulation directe en 3d d’un très grand nombre de cils. Nous l’avons donc appliquée au cas du transport mucociliaire dans les poumons. Cet outil numérique nous donne accès à des informations que l’on ne peut avoir par l’expérience, et permet de simuler des cas pathologiques comme par exemple une distribution éparse des cils. / Numerous mucous membranes inside the human body are covered with cilia which, by their coordinated movements, lead to a circulation of the layer of fluid coating the mucous membrane, which allows, for example, in the case of the internal wall of the bronchi, the evacuation of the impurities inspired outside the respiratory system.In this thesis, we integrate the effects of the cilia on the fluid, at the scale of the cilium. For this, we consider the incompressible Stokes equations. Due to the very small thickness of the cilia, the direct computation would request a time-varying mesh grading around the cilia. To avoid too prohibitive computational costs, we consider the asymptotic of a zero diameter cilium with an infinite velocity: the cilium is modelled by a lineic Dirac of force in source term. In order to ease the computations, the lineic Dirac of forces can be approached by a sum of punctual Dirac masses distributed along the cilium. Thus, by linearity, we have switched our initial problem with the Stokes problem with a punctual force in source term. Thus, we simplify the computations, but the final problem is more singular than the initial problem. The loss of regularity involves a deeper numerical analysis and the development of a new method to solve the problem.We have first studied a scalar version of this problem: Poisson problem with a Dirac right-hand side. The exact solution is singular, therefore the finite element solution has to be defined with caution. In this case, the convergence is not as good as in the regular case, and thus we focused on local error estimates. We have proved a quasi-optimal convergence in H1-norm (s ď 1) on a sub-domain which does not contain the singularity. Similar results have been shown for the Stokes problem too.In order to recover an optimal convergence on the whole domain, we have developped a numerical method to solve elliptic problems with a Dirac mass or a punctual force in source term. It is based on the standard finite element method and the explicit knowl- edge of the singularity of the exact solution. Given the positions of the cilia and their parametrisations, this method permits to compute in 3d a very high number of cilia. We have applied this to the study of the mucociliary transport in the lung. This numerical tool gives us information we do not have with the experimentations and pathologies can be computed and studied by this way, like for example a small number of cilia.

L’objectif principal de cette thèse est l’analyse des performances des réseaux LTE-A (Long Term Evolution- Advanced) au travers de la géométrie stochastique. L’analyse mathématique des réseaux cellulaires est un problème difficile, pour lesquels ils existent déjà un certain nombre de résultats mais qui demande encore des efforts et des contributions sur le long terme. L’utilisation de la géométrie aléatoire et des processus ponctuels de Poisson (PPP) s’est avérée être une approche permettant une modélisation pertinente des réseaux cellulaires et d’une complexité faible (tractable). Dans cette thèse, nous nous intéressons tout particulièrement à des modèles s’appuyant sur ces processus de Poisson : PPP-based abstraction. Nous développons un cadre mathématique qui permet le calcul de quantités reflétant les performances des réseaux LTE-A, tels que la probabilité d’erreur, la probabilité et le taux de couverture, pour plusieurs scénarios couvrant entre autres le sens montant et descendant. Nous considérons également des transmissions multi-antennes, des déploiements hétérogènes, et des systèmes de commande de puissance de la liaison montante. L’ensemble de ces propositions a été validé par un grand nombre de simulations. Le cadre mathématique développé dans cette thèse se veut général, et doit pouvoir s’appliquer à un nombre d’autres scénarios importants. L’intérêt de l’approche proposée est de permettre une évaluation des performances au travers de l’évaluation des formules, et permettent en conséquences d’éviter des simulations qui peuvent prendre énormément de temps en terme de développement ou d’exécution. / The main focus of this thesis is on performance analysis and system optimization of Long Term Evolution - Advanced (LTE-A) cellular networks by using stochastic geometry. Mathematical analysis of cellular networks is a long-lasting difficult problem. Modeling the network elements as points in a Poisson Point Process (PPP) has been proven to be a tractable yet accurate approach to the performance analysis in cellular networks, by leveraging the powerful mathematical tools such as stochastic geometry. In particular, relying on the PPP-based abstraction model, this thesis develops the mathematical frameworks to the computations of important performance measures such as error probability, coverage probability and average rate in several application scenarios in both uplink and downlink of LTE-A cellular networks, for example, multi-antenna transmissions, heterogeneous deployments, uplink power control schemes, etc. The mathematical frameworks developed in this thesis are general enough and the accuracy has been validated against extensive Monte Carlo simulations. Insights on performance trends and system optimization can be done by directly evaluating the formulas to avoid the time-consuming numerical simulations.

Cette thèse porte principalement sur la modélisation, l'évaluation des performances et l'optimisation au niveau système des réseaux cellulaires de nouvelle génération à l'aide de la géométrie stochastique. En plus, la technologie émergente des surfaces intelligentes reconfigurables (RISs) est étudiée pour l'application aux futurs réseaux sans fil. En particulier, reposant sur un modèle d’abstraction basé sur la loi de Poisson pour la distribution spatiale des nœuds et des points d’accès, cette thèse développe un ensemble de nouveaux cadres analytiques pour le calcul d’importantes métriques de performance, telles que la probabilité de couverture et l'efficacité spectrale potentielle, qui peuvent être utilisés pour l'analyse et l'optimisation au niveau système. Plus spécifiquement, une nouvelle méthodologie d'analyse pour l'analyse de réseaux cellulaires tridimensionnels est introduite et utilisée pour l'optimisation du système. Un nouveau problème d’allocation de ressources est formulé et résolu en combinant pour la première fois géométrie stochastique et programmation non linéaire mixte en nombres entiers. L'impact du déploiement de surfaces réfléchissantes intelligentes sur un réseau sans fil est quantifié à l'aide de processus ponctuels, et les avantages potentiels des RISs contre le relais sont étudiés à l'aide de simulations numériques. / The main focus of this thesis is on modeling, performance evaluation and system-level optimization of next-generation cellular networks by using stochastic geometry. In addition, the emerging technology of Reconfigurable Intelligent Surfaces (RISs) is investigated for application to future wireless networks. In particular, relying on a Poisson-based abstraction model for the spatial distribution of nodes and access points, this thesis develops a set of new analytical frameworks for the computation of important performance metrics, such as the coverage probability and potential spectral efficiency, which can be used for system-level analysis and optimization. More specifically, a new analytical methodology for the analysis of three-dimensional cellular networks is introduced and employed for system optimization. A novel resource allocation problem is formulated and solved by jointly combining for the first time stochastic geometry and mixed-integer non-linear programming. The impact of deploying intelligent reflecting surfaces throughout a wireless network is quantified with the aid of line point processes, and the potential benefits of RISs against relaying are investigated with the aid of numerical simulations.

The dissertation suggests a generalized version of Tikhonov regularization and analyzes its properties. The focus is on convergence rates theory and an extensive example for regularization with Poisson distributed data is given.