Research Doctorate - Doctor of Philosophy (PhD) / Interval estimation of the Binomial parameter è, representing the true probability of a success, is a problem of long standing in statistical inference. The landmark work is by Bayes (1763) who applied the uniform prior to derive the Beta posterior that is the normalised Binomial likelihood function. It is not well known that Bayes favoured this ‘noninformative’ prior as a result of considering the observable random variable x as opposed to the unknown parameter è, which is an important difference. In this thesis we develop additional arguments in favour of the uniform prior for estimation of è. We start by describing the frequentist and Bayesian approaches to interval estimation. It is well known that for common continuous models, while different in interpretation, frequentist and Bayesian intervals are often identical, which is directly related to the existence of a pivotal quantity. The Binomial model, and its Poisson sister also, lack a pivotal quantity, despite having sufficient statistics. Lack of a pivotal quantity is the reason why there is no consensus on one particular estimation method, more so than its discreteness: frequentist (unconditional) coverage depends on è. Exact methods guarantee minimum coverage to be at least equal to nominal and approximate methods aim for mean coverage to be close to nominal. We agree with what seems like the majority of frequentists, that exact methods are too conservative in practice, and show additional undesirable properties. This includes more recent ‘short’ exact intervals. We argue that Bayesian intervals based on noninformative priors are preferable to the family of frequentist approximate intervals, some of which are wider than exact intervals for particular data values. A particular property of the interval based on the uniform prior is that its mean coverage is exactly equal to nominal. However, once committed to the Bayesian approach there is no denying that the current preferred choice, by ‘objective’ Bayesians, is the U-shaped Jeffreys prior which results from various methods aimed at finding noninformative priors. The most successful such method seems to be reference analysis which has led to sensible priors in previously unsolved problems, concerning multiparameter models that include ‘nuisance’ parameters. However, we argue that there is a class of models for which the Jeffreys/reference prior may be suboptimal and that in the case of the Binomial distribution the requirement of a uniform prior predictive distribution leads to a more reasonable ‘consensus’ prior.

Classical and Bayesian methods for interval estimation of the ratio of two independent Poisson rates are examined and compared in terms of their exact coverage properties. Two methods to determine sampling effort requirements are derived.

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.