Density estimation for some semiparametric models
Open Access
- Author:
- Hernandez Bejarano, Manuel
- Graduate Program:
- Statistics
- Degree:
- Doctor of Philosophy
- Document Type:
- Dissertation
- Date of Defense:
- February 23, 2024
- Committee Members:
- Bing Li, Professor in Charge/Director of Graduate Studies
Lingzhou Xue, Major Field Member
Andres Aradillas-Lopez, Outside Unit & Field Member
Zhibiao Zhao, Chair & Dissertation Advisor
Runze Li, Major Field Member - Keywords:
- Density estimation
Functional convergence
Nonparametric kernel den- sity estimator
Nonparametric regression
Specification testing
Density estimation
Functional convergence
Nonparametric kernel density estimator
Nonparametric regression
Specification testing
ARCH models
Quantile regression - Abstract:
- The classical nonparametric kernel density estimator has been widely used to estimate the marginal density of a variable of interest in fields such as finance and economics. However, it has several limitations, including a slow convergence rate, which becomes particularly problematic in small sample sizes. This dissertation document is concerned with studying more efficient density estimations for the marginal density of two important semiparametric models. We show that the proposed estimators exhibit appealing properties that are absent in the classical estimator. In the first project, chapter 2, motivated by the slow convergence rate of classical nonparametric kernel density estimator, we study more efficient density estimation and density derivative estimation for the marginal density of nonparametric regression models. We show that in the presence of unknown nonparametric regression function, the proposed density and density derivative estimators can achieve parametric convergence rate, √n, and possess several appealing properties which the classical estimator lacks. Also, in the absence of nonparametric regression function, when the noise is normally distributed, the proposed method performs as well as if we have known the model and estimated the density using the maximum likelihood method. Based on the proposed density estimator, we further propose a more powerful density-based specification test for the nonparametric regression function. Our extensive numerical studies show that the proposed density estimator, density derivative estimator, and specification test significantly outperform existing ones. In the second project, chapter 3, we study a more efficient density estimation for the stationary density of nonparametric autoregressive conditional heteroscedasticity (NARCH) models. These models are important tools in analyzing time series, specifically in economic and financial applications where the goal is modeling and understanding the volatility of the statistical data since this volatility appears to change over time and exhibit clustering. We demonstrate that in the presence of an unknown nonparametric variance structure, we can establish the root-n consistency of the proposed density estimator, improving this way the widely used nonparametric kernel density estimator whose rate of convergence is inferior. A numerical study confirms the results. The density estimator is applied to the S&P 500 Index data. Finally, we showcase a practical implementation of the proposed density estimator in quantile regression. Specifically, we propose to get a more accurate estimate of the limiting variance of the estimated coefficients in a quantile regression model whose errors follow a nonparametric autoregressive conditional heteroscedastic structure. We perform a simulation study, which shows that using the new density estimator leads to a more accurate estimation of this asymptotic variance compared to the results obtained using the classical density estimator. To illustrate the application of this methodology in estimating the asymptotic variance, we apply it to the monthly inflation rate of the United States. Finally, Chapter 4 summarizes the main conclusions of the projects outlined in this document, as well as two potential avenues for future research in density estimation in the context of time series.