Estimation and testing under shape constraints

Abstract

This thesis consists of three projects, the common thread to all of which is using shape-restricted densities in inference problems. In the first project, we revisit the problem of estimating the center of symmetry of an unknown symmetric density. This problem dates back to Stone (1975), Van Eden (1970), and Sacks (1975), who constructed adaptive estimators relying on tuning parameters. Our third project, which aims to compare the outcomes from two vaccine trials, focuses on developing methodologies for testing stochastic dominance and estimating the Hellinger distance between densities. In both of these projects, we impose an additional shape restriction of either log-concavity or unimodality on the underlying densities. We show that, in both cases, the introduction of shape restrictions lead to simpler inference procedures, relying on either only one tuning parameter or none. My other project introduces a new shape-constrained class of distribution functions on the real line, the bi-s*-concave} class, which, in parallel to the results of Dumbgen et al. (2017) extends the class of s-concave densities to a class including possibly multi-modal densities.