UC Davis

In this dissertation, we are tackling two estimation problems using methods with geometric arguments heavily involved. The first part of this dissertation is focused on theoretical analysis of a type of local convex hull (LoCoH) estimator of the density support, which includes the r-ball LoCoH estimator and the kNN LoCoH estimator. We present two finite-sample concentration bounds for both estimators and discuss how to use these bounds to study the large sample behaviors and the performance of these estimators in high dimensions. The second part of this dissertation is focused on establishing a generalized switching relation, which states that the behavior of the nonparametric maximum likelihood estimators (NPMLEs) of density functions under order constraints can be controlled by the behavior of the empirical measure. Some examples are shown to illustrate how to use this relation to derive asymptotic results.