Date of Award

Spring 1-1-2015

Document Type


Degree Name

Doctor of Philosophy (PhD)



First Advisor

Carlos Martins-Filho

Second Advisor

Kairat Mynbaev

Third Advisor

Donald Waldman

Fourth Advisor

Xiaodong Liu

Fifth Advisor

Scott Savage


In the first chapter of this thesis, a class of local constant kernel estimators for a regression in Besov spaces is developed based on a novel set of kernels provided by Mynbaev and Martins-Filho (2010). The proposed class of local constant estimators includes the Nadaraya-Watson estimator. I show that bias reduction for the estimators in the class can be achieved without the potential negativity of the underlying estimated densities. Our estimators have faster uniform convergence rates than the Nadaraya-Watson estimator. I establish consistency and asymptotic normality of the estimators in the class. These results have been established without using higher-order kernels and imposing less restrictive conditions on the true density and the regression. A Monte Carlo study is provided to illustrate the finite sample performance of the estimators.

In the second chapter of this thesis, I propose a family of estimators for a measure of polarization via a kernel-based density estimator provided by Mynbaev and Martins-Filho (2010), as well as a distribution function estimator based on integration of the estimated density. The existing estimator for polarization measure proposed by Duclos et al. (2004) is based on the empirical distribution that suffers from lack of smoothness. I modified their work by using both the density estimators proposed by Mynbaev and Martins-Filho (2010) and the integration of the estimated density. I show that a class of estimators for the distribution function is asymptotically unbiased and consistent. In addition, I establish that a class of estimators for polarization measure is asymptotically unbiased. Finally, I study the behavior of the estimators using a Monte Carlo simulation.