Ce mémoire traite d'une méthode bayésienne, analysée par Marchand et Strawderman (2013), pour la construction d'intervalles bayésiens pour des modèles de densités continues avec contrainte sur l'espace des paramètres Θ. Notamment, on obtiendra une classe d'intervalles bayésiens Iπ0,α(.), associés à la troncature d'une loi a priori non informative π0 et générés par une fonction de distribution α(.), avec une probabilité de recouvrement bornée inférieurement par 1-α/1+α. Cette classe inclut la procédure HPD donnée par Marchand et Strawderman (2006) dans le cas où la densité sous-jacente d'un pivot est symétrique. Plusieurs exemples y illustrent la théorie étudiée. Finalement, on présentera de nouveaux résultats pour la probabilité de recouvrement des intervalles bayésiens appartenant à la classe étudiée pour des densités log-concaves. Ces résultats établissent la borne inférieure à 1- 3α/2 et généralisent les résultats de Marchand et al.(2008) tenant sous une hypothèse de symétrie.

Different universal methods (also called automatic or black-box methods) have been suggested to sample from univariate log-concave distributions. The description of a suitable universal generator for multivariate distributions in arbitrary dimensions has not been published up to now. The new algorithm is based on the method of transformed density rejection. To construct a hat function for the rejection algorithm the multivariate density is tranformed by a proper transformation T into a concave function (in the case of log-concave density T(x) = log(x).) Then it is possible to construct a dominating function by taking the minimum of several tangent hyperplanes which are transformed back by $T^(-1)$ into the original scale. The domains of different pieces of the hat function are polyhedra in the multivariate case. Although this method can be shown to work, it is too slow and complicated in higher dimensions. In this paper we split the $R^n$ into simple cones. The hat function is constructed piecewise on each of the cones by tangent hyperplanes. The resulting function is not continuous any more and the rejection constant is bounded from below but the setup and the generation remains quite fast in higher dimensions, e.g. n=8. The paper describes the details how this main idea can be used to construct algorithm TDRMV that generates random tuples from multivariate log-concave distribution with a computable density. Although the developed algorithm is not a real black box method it is adjustable for a large class of log-concave densities. (author's abstract) / Series: Preprint Series / Department of Applied Statistics and Data Processing

A rejection algorithm, called ``transformed density rejection", is presented. It uses a new method for constructing simple hat functions for a unimodal density $f$. It is based on the idea of transforming $f$ with a suitable transformation $T$ such that $T(f(x))$ is concave. The hat function is then constructed by taking the pointwise minimum of tangents which are transformed back to the original scale. The resulting algorithm works very well for a large class of distributions and is fast. The method is also extended to the two- and multidimensional case. (author's abstract) / Series: Preprint Series / Department of Applied Statistics and Data Processing

Ce m??moire traite d'une m??thode bay??sienne, analys??e par Marchand et Strawderman (2013), pour la construction d'intervalles bay??siens pour des mod??les de densit??s continues avec contrainte sur l'espace des param??tres ??. Notamment, on obtiendra une classe d'intervalles bay??siens I??0,??(.), associ??s ?? la troncature d'une loi a priori non informative ??0 et g??n??r??s par une fonction de distribution ??(.), avec une probabilit?? de recouvrement born??e inf??rieurement par 1-??/1+??. Cette classe inclut la proc??dure HPD donn??e par Marchand et Strawderman (2006) dans le cas o?? la densit?? sous-jacente d'un pivot est sym??trique. Plusieurs exemples y illustrent la th??orie ??tudi??e. Finalement, on pr??sentera de nouveaux r??sultats pour la probabilit?? de recouvrement des intervalles bay??siens appartenant ?? la classe ??tudi??e pour des densit??s log-concaves. Ces r??sultats ??tablissent la borne inf??rieure ?? 1- 3??/2 et g??n??ralisent les r??sultats de Marchand et al.(2008) tenant sous une hypoth??se de sym??trie.

This thesis deals with the problem of estimating statistical distributions from data. In the first part, the method of moments is used in combination with computational algebraic techniques in order to estimate parameters coming from local Dirac mixtures and their convolutions. The second part focuses on the nonparametric setting, in particular on combinatorial and algebraic aspects of the estimation of log-concave distributions.

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This thesis is based on three papers on selected topics in Asymptotic Geometric Analysis. The first paper is about the volume of high-dimensional random polytopes; in particular, on polytopes generated by Gaussian random vectors. We consider the question of how many random vertices (or facets) should be sampled in order for such a polytope to capture significant volume. Various criteria for what exactly it means to capture significant volume are discussed. We also study similar problems for random polytopes generated by points on the Euclidean sphere. The second paper is about volume distribution in convex bodies. The first main result is about convex bodies that are (i) symmetric with respect to each of the coordinate hyperplanes and (ii) in isotropic position. We prove that most linear functionals acting on such bodies exhibit super-Gaussian tail-decay. Using known facts about the mean-width of such bodies, we then deduce strong lower bounds for the volume of certain caps. We also prove a converse statement. Namely, if an arbitrary isotropic convex body (not necessarily satisfying the symmetry assumption (i)) exhibits similar cap-behavior, then one can bound its mean-width. The third paper is about random polytopes generated by sampling points according to multiple log-concave probability measures. We prove related estimates for random determinants and give applications to several geometric inequalities; these include estimates on the volume-radius of random zonotopes and Hadamard's inequality for random matrices. / Mathematics

This paper studies minimally-supported D-optimal designs for polynomial regression model with logarithmically concave (log-concave) weight functions. Many commonly used weight functions in the design literature are log-concave. We show that the determinant of information matrix of minimally-supported design is a log-concave function of ordered support points and the D-optimal design is unique. Therefore, the numerically D-optimal designs can be determined e¡Óciently by standard constrained concave programming algorithms.

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A rejection algorithm - called transformed density rejection - that uses a new method for constructing simple hat functions for an unimodal, bounded density $f$ is introduced. It is based on the idea to transform $f$ with a suitable transformation $T$ such that $T(f(x))$ is concave. $f$ is then called $T$-concave and tangents of $T(f(x))$ in the mode and in a point on the left and right side are used to construct a hat function with table-mountain shape. It is possible to give conditions for the optimal choice of these points of contact. With $T=-1/\sqrt(x)$ the method can be used to construct a universal algorithm that is applicable to a large class of unimodal distributions including the normal, beta, gamma and t-distribution. (author's abstract) / Series: Preprint Series / Department of Applied Statistics and Data Processing