Abstract:

Natural shape constraints can be used for semi/nonparametric estimation and inference in a wide variety of problems . In the talk we will consider two such examples.

First we consider a two-component mixture model with one known component. Given independent and  identically distributed (i.i.d.)~data from the mixture model, we develop methods for estimating the mixing proportion and the unknown distribution nonparametrically.  We establish the consistency of our estimators. We find the rate of convergence and asymptotic limit of the estimator for the mixing proportion. Completely automated distribution-free honest finite sample lower confidence bounds are developed for the mixing proportion. The identifiability of the model, and the estimation of the density of the unknown distribution are also addressed.

In the second part of the talk, we consider the single index regression model with an unknown convex link function. We propose a Lipschitz constrained least squares estimator (LLSE) for the nonparametric link function and the unknown finite dimensional parameter.  We prove the consistency and find the rates of convergence of the LLSE given i.i.d.~data. Furthermore, we establish root-n rate of convergence and semiparametric efficiency of the parametric component under mild assumptions. Moreover, the LLSE readily yields asymptotic confidence sets for the finite dimensional parameter.