# Efficient parameter estimation methods using quantile regression in heteroscedastic models

Open Access
Author:
Xu, Zhanxiong
Statistics
Degree:
Doctor of Philosophy
Document Type:
Dissertation
Date of Defense:
April 14, 2017
Committee Members:
The quantile regression method, first introduced by \citet{koenker1978regression}, provides a comprehensive toolkit of performing statistical inference for a class of statistical models and has become an important surrogate for the conventional least squares method. Specifically, quantile regression offers several versatile approaches to produce highly efficient estimates, regardless whether the error distribution is homoscedastic or not. This dissertation is concerned with developing some efficient estimation methods for both the regression parameter and the dispersion parameter under the parametric nonlinear heteroscedastic model. The proposed methods have their roots in quantile regression and rely heavily on large-sample properties of the estimates. In Chapter 2, we estimate the parameters by solving the double-weighted composite quantile regression (DWCQR)'' optimization problem. We establish central limit theorems for both estimates, based on which we recommend an objective way of choosing the optimal weights for both the quantile losses and the heteroscedasticity. It is shown by theoretical calculation that the resulting estimates are typically more efficient than those obtained from other methods, and their asymptotic variances converge to the Cram\'{e}r-Rao lower bounds as the number of quantile positions tends to infinity. An adaptive estimation procedure is reported at the end of this chapter. The computational aspects of the DWCQR problem are discussed in Chapter 3. Although the DWCQR problem, in general, does not admit numerical solutions that are guaranteed to converge, we attempted to provide an algorithm that combines the MM algorithm (\citet{hunter2000quantile}) and the linear programming. The proposed MMLP algorithm overall works well and successfully confirms the nice theoretical properties of the DWCQR estimates using the optimal weights. The Monte Carlo study demonstrates that the DWCQR method outperforms the conventional estimation methods for the models under investigation. In Chapter 4, for simplicity, we restrict the regression function to be linear and consider an alternative efficient estimation approach, which is based on a preliminary estimate $\hat{\alpha}_n$ of the dispersion parameter. We first derive the Bahadur representation of the regression quantile $\hat{\beta}(\tau)$ for fixed $\tau$. It is then interesting to note that the effect of the $\hat{\alpha}_n$ propagates in the asymptotic representation of $\hat{\beta}(\tau)$. Such asymptotic bias brought by $\hat{\alpha}_n$ can be eliminated by averaging regression quantiles across different quantile positions with a set of carefully chosen weights. In the meantime, it can be shown that these weights can be simultaneously adjusted so that the resulting estimate is also asymptotically efficient. The chapter is concluded by Monte Carlo studies.