Convex analysis methods in shape constrained estimation

We discuss applications of convex analysis to shape constrained density estimation. The dissertation consists of three parts.;In the first part we introduce convex transformed densities as a multivariate generalization of known classes of densities defined by shape constraints based on convexity. We study the properties of the nonparametric maximum likelihood estimator of a convex-transformed density in several dimensions and prove basic properties: existence and consistency.;In the second part we establish the local rates of convergence for the MLE of a power-convex convex density in one dimension. Some of the results describing the local behavior of the MLE hold in a general case of multivariate convex-transformed densities.;The third part includes results about the behavior of the MLE for mixture models. We provide upper and lower stochastic bounds for a wide range of scale mixture models which is an important step towards establishing global rates of convergence of the MLE. We also give a proof of the uniqueness of the k-monotone MLE.