O FBST (Full Bayesian Significance Test) é um instrumento desenvolvido por Pereira e Stern (1999) com o objetivo de apresentar uma alternativa bayesiana aos testes de hipóteses precisas. Desde sua introdução, o FBST se mostrou uma ferramenta muito útil para a solução de problemas para os quais não havia soluções freqüentistas. Esse teste, contudo, depende de que a amostra seja coletada uma única vez, após o que a distribuição a posteriori dos parâmetros é obtida e a medida de evidência, calculada. Ensejadas por esse aspecto, são apresentadas abordagens analíticas e computacionais para a extensão do FBST ao contexto de decisão seqüencial (DeGroot, 2004). É apresentado e analisado um algoritmo para a execução do FBST Seqüencial, bem como o código-fonte de um software baseado nesse algoritmo. / FBST (Full Bayesian Significance Test) is a tool developed by Pereira and Stern (1999), to show a bayesian alternative to the tests of precise hypotheses. Since its introduction, FBST has shown to be a very useful tool to solve problems to which there were no frequentist solutions. This test, however, needs that the sample be collected just one time and, after this, the parameters posterior distribution is obtained and the evidence measure, computed. Suggested by this feature, there are presented analytic and computational approaches to the extension of the FBST to the sequential decision context (DeGroot, 2004). It is presented and analyzed an algorithm to execute the Sequential FBST, as well as the source code of a software based on this algorithm.

O FBST (Full Bayesian Significance Test) é um instrumento desenvolvido por Pereira e Stern (1999) com o objetivo de apresentar uma alternativa bayesiana aos testes de hipóteses precisas. Desde sua introdução, o FBST se mostrou uma ferramenta muito útil para a solução de problemas para os quais não havia soluções freqüentistas. Esse teste, contudo, depende de que a amostra seja coletada uma única vez, após o que a distribuição a posteriori dos parâmetros é obtida e a medida de evidência, calculada. Ensejadas por esse aspecto, são apresentadas abordagens analíticas e computacionais para a extensão do FBST ao contexto de decisão seqüencial (DeGroot, 2004). É apresentado e analisado um algoritmo para a execução do FBST Seqüencial, bem como o código-fonte de um software baseado nesse algoritmo. / FBST (Full Bayesian Significance Test) is a tool developed by Pereira and Stern (1999), to show a bayesian alternative to the tests of precise hypotheses. Since its introduction, FBST has shown to be a very useful tool to solve problems to which there were no frequentist solutions. This test, however, needs that the sample be collected just one time and, after this, the parameters posterior distribution is obtained and the evidence measure, computed. Suggested by this feature, there are presented analytic and computational approaches to the extension of the FBST to the sequential decision context (DeGroot, 2004). It is presented and analyzed an algorithm to execute the Sequential FBST, as well as the source code of a software based on this algorithm.

J'ai consacré mon travail de thèse à la recherche de quasars à haut redshift dans le sondage Canada France High-z Quasar Survey in the Near Infrared (CFHQSIR). L'objectif principal de ce programme est de découvrir des quasars de redshift de l'ordre de z $\sim$ 7 dans les champs larges du CFHTLS (CFHTLS Wide) en utilisant des images réalisées dans le proche infrarouge avec la caméra WIRCam installée au foyer du CFHT. J'ai tout d'abord effectué la calibration photométrique de l'ensemble des données de CFHQSIR. J'ai ensuite étudié la qualité des images ainsi que leur propriété de bruit et leur profondeur. J'ai dans un deuxième temps calculé le nombre de quasars qu'il serait possible de détecter avec CFHQSIR et en ai déduit les contraintes envisageables sur la fonction de luminosité des quasars à z $\sim$ 7. J'ai ensuite procédé à l'identification de candidats quasars parmi les dizaines de milliers de sources que comptent les 130 degrés carrés couvert par CFHQSIR. Des observations de suivi photométriques des candidats ont finalement permis de révéler une cinquantaine d’objets, dont une quinzaine a été sélectionnée pour des observations spectroscopiques au Very Large Telescope (VLT). Celles-ci n’ayant, pour la plupart, pas encore été realisées, la nature exacte de ces sources ne pourra être connue que dans les prochains mois. Pour finir, la dernière partie de ma thèse s’est focalisée sur l’étude de modèles statistiques bayésiens afin de compléter ma méthode de sélection de candidats quasars. Cette étude m'a finalement permis de vérifier que la majorité des objets retenus pour des observations spectroscopiques étaient effectivement les candidats les plus probables. / My PhD work is focused on the search for high-redshift quasars in the Canada France High-z Quasar Survey in the Near Infrared (CFHQSIR). The main scientific objective of this CFHT Large Program is to search for quasars at redshift z $\sim$ 7 with near-infrared images of the CFHTLS Wide fields acquired with the CFHT WIRCam camera. Firstly, I carried out the photometric calibration of the CFHQSIR images. I performed a detailed analysis of the CFHQSIR data by studying their quality, as well as their noise properties and their depths.Secondly, I computed the number of high-redshift quasars that could be detected with CFHQSIR and the corresponding constraints which could be put on the z $\sim$ 7 quasar luminosity function. Then, I proceeded to the identification of quasar candidates among the many thousands of sources in the 130 square degrees covered by CFHQSIR. Photometric follow-up observations of the candidates revealed about fifty objects, of which fifteen were chosen to be spectroscopically observed with the Very Large Telescope (VLT). Given that this spectroscopic follow-up is not yet completed, the exact nature of these sources will only be known in the coming months. To finish, I applied Bayesian model comparison to my sample in order to complete and consolidate my selection procedure. My candidates were finally classified according to their probability to be a high-redshift quasar. I verified that the majority of the most likely candidates were selected for spectroscopic observations.

Le but de cette thèse est d’étudier l’approximation d’a priori impropres par des suites d’a priori propres. Nous définissons un mode de convergence sur les mesures de Radon strictement positives pour lequel une suite de mesures de probabilité peut admettre une mesure impropre pour limite. Ce mode de convergence, que nous appelons convergence q-vague, est indépendant du modèle statistique. Il permet de comprendre l’origine du paradoxe de Jeffreys-Lindley. Ensuite, nous nous intéressons à l’estimation de la taille d’une population. Nous considérons le modèle du removal sampling. Nous établissons des conditions nécessaires et suffisantes sur un certain type d’a priori pour obtenir des estimateurs a posteriori bien définis. Enfin, nous montrons à l’aide de la convergence q-vague, que l’utilisation d’a priori vagues n’est pas adaptée car les estimateurs obtenus montrent une grande dépendance aux hyperparamètres. / The purpose of this thesis is to study the approximation of improper priors by proper priors. We define a convergence mode on the positive Radon measures for which a sequence of probability measures could converge to an improper limiting measure. This convergence mode, called q-vague convergence, is independant from the statistical model. It explains the origin of the Jeffreys-Lindley paradox. Then, we focus on the estimation of the size of a population. We consider the removal sampling model. We give necessary and sufficient conditions on the hyperparameters in order to have proper posterior distributions and well define estimate of abundance. In the light of the q-vague convergence, we show that the use of vague priors is not appropriate in removal sampling since the estimates obtained depend crucially on hyperparameters.