Nonparametric estimation of time-varying covariance matrix in a slowly changing vector random walk model

A new multivariate random walk model with slowly changing parameters is introduced and investigated in detail. Nonparametric estimation of local covariance matrix is proposed. The asymptotic distributions, including asymptotic biases, variances and covariances of the proposed estimators are obtained. The properties of the estimated value of a weighted sum of individual nonparametric estimators are also studied in detail. The integrated effect of the estimation errors from the estimation for the difference series to the integrated processes is derived. Practical relevance of the model and estimation is illustrated by application to several foreign exchange rates.Multivariate time series; slowly changing vector random walk; local covariance matrix; kernel estimation; asymptotic properties; forecasting

Smooth regression quantile estimation

In this thesis, attention will be mainly focused on the local linear kernel regression quantile estimation. Different estimators within this class have been proposed, developed asymptotically and applied to real applications. I include algorithmdesign and selection of smoothing parameters. Chapter 2 studies two estimators, first a single-kernel estimator based on "check function" and a bandwidth selection rule is proposed based on the asymptotic MSE of this estimator. Second a recursive double-kernel estimator which extends Fan et al's (1996) density estimator, and two algorithms are given for bandwidth selection. In Chapter 3, a comparison is carried out of local constant fitting and local linear fitting using MSEs of the estimates as a criterion. Chapter 4 gives a theoretical summary and a simulation study of local linear kernel estimation of conditional distribution function. This has a special interest in itself as well as being related to regression quantiles. In Chapter 5, a kernel-version method of LMS (Cole and Green, 1992) is considered. The method proposed, which is still a semi-parametric one, is based on a general idea of local linear kernel approach of log-likelihood model. Chapter 6 proposes a two-step method of smoothing regression quantiles called BPK. The method considered is based on the idea of combining k- NN method with Healy's et al (1988) partition rule, and correlated regression model are involved. In Chapter 7, methods of regression quantile estimation are compared for different underlying models and design densities in a simulation study. The ISE criterion of interior and boundary points is used as a basis for these comparisons. Three methods are recommended for quantile regression in practice, and they are double kernel method, LMS method and Box partition kernel method (BPK). In Chapter 8, attention is turned to a novel idea of local polynomial roughness penalty regression model, where a purely theoretical framework is considered

Bayesian adaptive lasso quantile regression

Recently, variable selection by penalized likelihood has attracted much research interest. In this paper, we propose adaptive Lasso quantile regression (BALQR) from a Bayesian perspective. The method extends the Bayesian Lasso quantile regression by allowing different penalization parameters for different regression coefficients. Inverse gamma prior distributions are placed on the penalty parameters. We treat the hyperparameters of the inverse gamma prior as unknowns and estimate them along with the other parameters. A Gibbs sampler is developed to simulate the parameters from the posterior distributions. Through simulation studies and analysis of a prostate cancer dataset, we compare the performance of the BALQR method proposed with six existing Bayesian and non-Bayesian methods. The simulation studies and the prostate cancer data analysis indicate that the BALQR method performs well in comparison to the other approaches

Modelling financial time series with SEMIFAR-GARCH model

A class of semiparametric fractional autoregressive GARCH models (SEMIFAR-GARCH), which includes deterministic trends, difference stationarity and stationarity with short- and long-range dependence, and heteroskedastic model errors, is very powerful for modelling financial time series. This paper discusses the model fitting, including an efficient algorithm and parameter estimation of GARCH error term. So that the model can be applied in practice. We then illustrate the model and estimation methods with a few of different finance data sets.Financial time series; GARCH model; SEMIFAR model; parameter estimation; kernel estimation; asymptotic property

Modelling financial time series with SEMIFAR-GARCH model

A class of semiparametric fractional autoregressive GARCH models (SEMIFARGARCH), which includes deterministic trends, difference stationarity and stationarity with short- and long-range dependence, and heteroskedastic model errors, is very powerful for modelling financial time series. This paper discusses the model fitting, including an efficient algorithm and parameter estimation of GARCH error term. So that the model can be applied in practice. We then illustrate the model and estimation methods with a few of different finance data sets.Financial time series, GARCH model, SEMIFAR model, parameter estimation, kernel estimation, asymptotic property.

Bayesian quantile regression: An application to the wage distribution in 1990s Britain

This paper illustrates application of Bayesian inference to quantile regression. Bayesian inference regards unknown parameters as random variables, and we describe an MCMC algorithm to estimate the posterior densities of quantile regression parameters. Parameter uncertainty is taken into account without relying on asymptotic approximations. Bayesian inference revealed effective in our application to the wage structure among working males in Britain between 1991 and 2001 using data from the British Household Panel Survey. Looking at different points along the conditional wage distribution uncovered important features of wage returns to education, experience and public sector employment that would be concealed by mean regression.quantile regression ; bayesian inference ; wage distribution ; MCMC

Bayesian Fractional Polynomial Approach to Quantile Regression and Variable Selection with Application in the Analysis of Blood Pressure among US Adults

Hypertension is a highly prevalent chronic medical condition and a strong risk factor for cardiovascular disease (CVD), as it accounts for more than $45\%$ of CVD. The relation between blood pressure (BP) and its risk factors cannot be explored clearly by standard linear models. Although the fractional polynomials (FPs) can act as a concise and accurate formula for examining smooth relationships between response and predictors, modelling conditional mean functions observes the partial view of a distribution of response variable, as the distributions of many response variables such as BP measures are typically skew. Then modelling 'average' BP may link to CVD but extremely high BP could explore CVD insight deeply and precisely. So, existing mean-based FP approaches for modelling the relationship between factors and BP cannot answer key questions in need. Conditional quantile functions with FPs provide a comprehensive relationship between the response variable and its predictors, such as median and extremely high BP measures that may be often required in practical data analysis generally. To the best of our knowledge, this is new in the literature. Therefore, in this paper, we employ Bayesian variable selection with quantile-dependent prior for the FP model to propose a Bayesian variable selection with parametric nonlinear quantile regression model. The objective is to examine a nonlinear relationship between BP measures and their risk factors across median and upper quantile levels using data extracted from the 2007-2008 National Health and Nutrition Examination Survey (NHANES). The variable selection in the model analysis identified that the nonlinear terms of continuous variables (body mass index, age), and categorical variables (ethnicity, gender and marital status) were selected as important predictors in the model across all quantile levels

A New Bayesian Huberised Regularisation and Beyond

Robust regression has attracted a great amount of attention in the literature recently, particularly for taking asymmetricity into account simultaneously and for high-dimensional analysis. However, the majority of research on the topics falls in frequentist approaches, which are not capable of full probabilistic uncertainty quantification. This paper first proposes a new Huberised-type of asymmetric loss function and its corresponding probability distribution which is shown to have the scale-mixture of normals. Then we introduce a new Bayesian Huberised regularisation for robust regression. A by-product of the research is that a new Bayesian Huberised regularised quantile regression is also derived. We further present their theoretical posterior properties. The robustness and effectiveness of the proposed models are demonstrated in the simulation studies and the real data analysis