Bayesian Nonparametric Estimation with Shape Constraints

Abstract

This thesis deals with a number of statistical problems where either censoring or shape-constraints play a role. These problems have mostly been treated from a frequentist statistical perspective. Over the past decades, the approach to statistics has gained popularity and this is the approach that is adopted in this thesis. We consider nonparametric statistical models, i.e. models indexed by a parameter that is not of finite dimension. For three different models we investigate the asymptotic properties of the posterior distribution under a frequentist setup. We derive either posterior consistency or posterior contraction rat es. Such results are relevant, as these provides a frequentist justification of using point estimators derived from the posterior. Besides theoretical results, we develop computational methods for obtaining draws from the posterior. Overall, this work is at the intersection of the research areas estimation under shape constraints and censoring, Bayesian nonparametrics and Bayesian computation.