Date of Award
Doctor of Philosophy in Analytic and Data Science
Statistics and Analytical Sciences
Dr. Mohammed Chowdhury
Dr. Lewis VanBrackle
Dr. Joe DeMaio
Dr. Xiao Huang
This dissertation develops and discusses several one-step and two-step smoothing methods of time variant nonparametric quantiles and time variant parameters from probability models. First, we investigate and develop nonparametric techniques for measuring extreme quantiles. The method involves aggregating data by an explanatory variable such as time and smoothing the resulting data with a nonparametric method like kernel, local polynomial or spline smoothing. We demonstrate both in application and simulation that this two-step procedure of quantile estimation is superior to the parametric quantile regression. We then develop a one-step method which combines the strength of maximum likelihood estimation with a local kernel function. This local maximum likelihood estimation is applied in both a discrete and continuous case of distribution, and we consider polynomial expansions of the unknown parameter in each case. In the continuous case, we choose a distribution with two parameters and iteratively solve for each to smooth the data. Results indicate that the one-step procedure can yield improvement over the corresponding two-step methods mentioned previously in both application cases and simulation exercises. We also explore nonparametric techniques for estimating volatility of financial data. We develop a residual based method for estimating the conditional variance function using local composite quantile regression, and compare this to using local least squares regression. These methods are applied on the asset returns for many individual firms, with promising results in favor of local composite quantile regression. Comparisons of these nonparametric techniques in forecasting also indicate some improvement over using a traditional autoregressive model for heteroscedastic data.