Title: Semiparametric inference with shape constraints
Description: This thesis deals with estimation and inference in two semiparametric problems: a two-component mixture model and a single index regression model. For the two-component mixture model, we assume that the distribution of one component is known and develop methods for estimating the mixing proportion and the unknown distribution using ideas from shape restricted function estimation. We establish the consistency of our estimators. We find the rate of convergence and the asymptotic limit of our estimator for the mixing proportion. Furthermore, we develop a completely automated distribution-free honest finite sample lower confidence bound for the mixing proportion. We compare the proposed estimators, which are easily implementable, with some of the existing procedures through simulation studies and analyse two data sets, one arising from an application in astronomy and the other from a microarray experiment. For the single index model, we consider estimation of the unknown link function and the finite dimensional index parameter.

We study the problem when the true link function is assumed to be: (1) smooth or (2) convex. When the link function is just assumed to be smooth, in contrast to standard kernel based methods, we use smoothing splines to estimate the link function. We prove the consistency and find the rates of convergence of the proposed estimators. We establish root-n-rate of convergence and the semiparametric efficiency of the parametric component under mild assumptions. When the link function is assumed to be convex, we propose a shape constrained penalized least squares estimator and a Lipschitz constrained least squares estimator for the unknown quantities. We prove the consistency and find the rates of convergence for both estimators. For the shape constrained penalized least squares estimator, we establish root-n-rate of convergence and the semiparametric efficiency of the parametric component under mild assumptions and conjecture that the parametric component of the Lipschitz constrained least squares estimator is semiparametrically efficient.

We develop the R package "simest'' that can be used (to compute the proposed estimators) even for moderately large dimensions.