Summary |
This thesis discusses nonparametric estimations in models with decreasing densities from a Bayesian viewpoint. The decreasing density is modeled as a scale mixture of uniforms. From a Bayes estimate of the unknown density, which is a predictive density of a sum over all partitions, we derive the predictive density as a sum over a class of vectors of special forms, called s-paths, through an approach that emphasizes statistical structure of the models. A correspondence between partitions and s-paths is defined. Then, given the posterior distribution of partitions, joint statistical structure between partitions and s-paths is described through a (marginal) posterior distribution of s-paths and a conditional distribution of a partition given a s-path, which turns out to be a discrete uniform distribution related to the correspondence defined. Since the predictive density of a sum over s-paths is a result of Rao-Blackwell theorem from the one related to partitions, s-paths should be adopted instead of partitions in this model to produce less variable estimates. To order to exploit the predictive density of a sum over all s-paths, this thesis introduces two path samplers based on the posterior distribution of s-paths. These two sampling methods for s-paths differ from other existing methods [Brunner and Lo (1989) and Brunner (1995)], in which the proposed algorithms draw variates of s-path from the desired posterior distribution. Numerical results based on the two path samplers are given, showing that the two path samplers give reasonable approximations. Two other existing methods, one based on "missing values" and the other based on partitions, are selected for the purpose of comparison. It turns out that the second introduced path sampler provides better estimations compared with the two existing methods. Estimations in a reliability model with a decreasing hazard rate are discussed in an analogous manner. A sequential importance sampler for drawing s-paths is proposed. |