DOI: 10.4225/03/58b3b55858126
title: Bayesian sampling for smoothing parameter estimation

description: Kernel density estimation is one of the most important techniques for understanding the distributional properties of data. It is understood that the effectiveness of such approach depends on the choice of a kernel function and the choice of a smoothing parameter (bandwidth). This thesis has undertaken some important topics in bandwidth selection for kernel density estimation for data that behave in various nature. The first issue evolves around selecting appropriate bandwidth given the characteristics of the local data in multivariate setting. In Chapter 3, the study proposes a kernel density estimator with tail-adaptive bandwidths. The study derives posterior of bandwidth parameters based on the Kullback-Leibler information and presented an MCMC sampling algorithm to estimate bandwidths. The Monte Carlo simulation study shows that the kernel density estimator with tail-adaptive bandwidths estimated through the proposed  sampling algorithm outperforms its competitor. The tail-adaptive kernel density estimator is applied to the estimation of bivariate density of the paired daily returns of the Australian Ordinary index and S&P 500 index during the period of global financial crisis. The results show that this estimator could capture richer dynamics in the tail area than the density estimator with a global bandwidth estimated through the normal reference rule and a Bayesian sampling algorithm.  

The second research project investigates bandwidth selection for multimodal distributions or data that exhibits clustering behaviours. Chapter 4 proposes a cluster-adaptive bandwidth kernel density estimator for data with multimodality. This method employs a clustering algorithm to assign a different bandwidth to each cluster identified in the data set. The study derives a posterior of bandwidth parameters based on the Kullback-Leibler information and presented an MCMC sampling algorithm to estimate bandwidths. The Monte Carlo simulation study shows that when the underlying density is a mixture of normals, the kernel density estimator with cluster-adaptive bandwidths estimated through the proposed  sampling algorithm outperforms its competitor. When the underlying densities are fat-tailed, the combined approach of tail- and cluster-adaptive density estimator performs the best. In an empirical study, bandwidth matrices are estimated for the cluster-adaptive kernel density estimator of eruption duration and waiting time to the next eruption collected from Old Faithful greyer, which is often analysed due to its clustering nature. The results again shows clear advantage of the proposed cluster-adaptive kernel density estimator over traditional approaches. 

The third topic extends the Bayesian bandwidth selection method to volatility models of financial asset return series. The study is motivated by the fact that only limited attention in the literature has been invested on the estimation of nonparametric nonlinear type of volatility models through a Bayesian approach. Chapter 5 presents a new volatility model called the semiparametic nonlinear volatility (SNV) model. Based on financial return series of major stock indices in the world, the performance of the proposed volatility model against the competing models are examined in both in-sample and out-of-sample periods. The proposed model and the Bayesian estimation method show strong and convincing performance results. The study also evaluates the empirical value-at-risk (VaR) performance of the competing models. The proposed volatility model shows the best performance in most cases.