Local Signal Detection on Irregular Domains with Spatially Varying Coefficient Model

In areas such as spatial analysis and time series analysis, it is essential to understand and quantify spatial or temporal heterogeneity. In this dissertation, we focus on a spatially varying coefficient model, in which spatial heterogeneity is accommodated by allowing the regression coefficients to vary in a given spatial domain. We propose a model selection method for spatially varying coefficient models using penalized bivariate splines. It uses bivariate splines defined on triangulation to approximate nonparametric varying coefficient functions and minimizes the sum of squared errors with local penalty on $L_2$ norms of spline coefficients for each triangle. Our method partitions the region of interest using triangulation and provides efficient approximation of irregular domains. In addition, we propose an efficient algorithm to obtain the proposed estimator using the local quadratic approximation. We also establish the consistency of estimated nonparametric coefficient functions and the estimated null regions. Moreover, we develop model confidence regions as the inference tool to quantify the uncertainty of the estimated null regions. The numerical performance of the proposed method is evaluated in both simulation case and real data analysis

A selective overview of nonparametric methods in financial econometrics

This paper gives a brief overview on the nonparametric techniques that are useful for financial econometric problems. The problems include estimation and inferences of instantaneous returns and volatility functions of time-homogeneous and time-dependent diffusion processes, and estimation of transition densities and state price densities. We first briefly describe the problems and then outline main techniques and main results. Some useful probabilistic aspects of diffusion processes are also briefly summarized to facilitate our presentation and applications.Comment: 32 pages include 7 figure

Quantile regression in partially linear varying coefficient models

Semiparametric models are often considered for analyzing longitudinal data for a good balance between flexibility and parsimony. In this paper, we study a class of marginal partially linear quantile models with possibly varying coefficients. The functional coefficients are estimated by basis function approximations. The estimation procedure is easy to implement, and it requires no specification of the error distributions. The asymptotic properties of the proposed estimators are established for the varying coefficients as well as for the constant coefficients. We develop rank score tests for hypotheses on the coefficients, including the hypotheses on the constancy of a subset of the varying coefficients. Hypothesis testing of this type is theoretically challenging, as the dimensions of the parameter spaces under both the null and the alternative hypotheses are growing with the sample size. We assess the finite sample performance of the proposed method by Monte Carlo simulation studies, and demonstrate its value by the analysis of an AIDS data set, where the modeling of quantiles provides more comprehensive information than the usual least squares approach.Comment: Published in at http://dx.doi.org/10.1214/09-AOS695 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Estimation of constant and time-varying dynamic parameters of HIV infection in a nonlinear differential equation model

Modeling viral dynamics in HIV/AIDS studies has resulted in a deep understanding of pathogenesis of HIV infection from which novel antiviral treatment guidance and strategies have been derived. Viral dynamics models based on nonlinear differential equations have been proposed and well developed over the past few decades. However, it is quite challenging to use experimental or clinical data to estimate the unknown parameters (both constant and time-varying parameters) in complex nonlinear differential equation models. Therefore, investigators usually fix some parameter values, from the literature or by experience, to obtain only parameter estimates of interest from clinical or experimental data. However, when such prior information is not available, it is desirable to determine all the parameter estimates from data. In this paper we intend to combine the newly developed approaches, a multi-stage smoothing-based (MSSB) method and the spline-enhanced nonlinear least squares (SNLS) approach, to estimate all HIV viral dynamic parameters in a nonlinear differential equation model. In particular, to the best of our knowledge, this is the first attempt to propose a comparatively thorough procedure, accounting for both efficiency and accuracy, to rigorously estimate all key kinetic parameters in a nonlinear differential equation model of HIV dynamics from clinical data. These parameters include the proliferation rate and death rate of uninfected HIV-targeted cells, the average number of virions produced by an infected cell, and the infection rate which is related to the antiviral treatment effect and is time-varying. To validate the estimation methods, we verified the identifiability of the HIV viral dynamic model and performed simulation studies.Comment: Published in at http://dx.doi.org/10.1214/09-AOAS290 the Annals of Applied Statistics (http://www.imstat.org/aoas/) by the Institute of Mathematical Statistics (http://www.imstat.org

Large-scale Heteroscedastic Regression via Gaussian Process

Heteroscedastic regression considering the varying noises among observations has many applications in the fields like machine learning and statistics. Here we focus on the heteroscedastic Gaussian process (HGP) regression which integrates the latent function and the noise function together in a unified non-parametric Bayesian framework. Though showing remarkable performance, HGP suffers from the cubic time complexity, which strictly limits its application to big data. To improve the scalability, we first develop a variational sparse inference algorithm, named VSHGP, to handle large-scale datasets. Furthermore, two variants are developed to improve the scalability and capability of VSHGP. The first is stochastic VSHGP (SVSHGP) which derives a factorized evidence lower bound, thus enhancing efficient stochastic variational inference. The second is distributed VSHGP (DVSHGP) which (i) follows the Bayesian committee machine formalism to distribute computations over multiple local VSHGP experts with many inducing points; and (ii) adopts hybrid parameters for experts to guard against over-fitting and capture local variety. The superiority of DVSHGP and SVSHGP as compared to existing scalable heteroscedastic/homoscedastic GPs is then extensively verified on various datasets.Comment: 14 pages, 15 figure