A profile-type smoothed score function for a varying coefficient partially linear model

The varying coefficient partially linear model is considered in this paper. When the plug-in estimators of coefficient functions are used, the resulting smoothing score function becomes biased due to the slow convergence rate of nonparametric estimations. To reduce the bias of the resulting smoothing score function, a profile-type smoothed score function is proposed to draw inferences on the parameters of interest without using the quasi-likelihood framework, the least favorable curve, a higher order kernel or under-smoothing. The resulting profile-type statistic is still asymptotically Chi-squared under some regularity conditions. The results are then used to construct confidence regions for the parameters of interest. A simulation study is carried out to assess the performance of the proposed method and to compare it with the profile least-squares method. A real dataset is analyzed for illustration.     

Robust and efficient variable selection for semiparametric partially linear varying coefficient model based on modal regression

Semiparametric partially linear varying coefficient models (SPLVCM) are frequently used in statistical modeling. With high-dimensional covariates both in parametric and nonparametric part for SPLVCM, sparse modeling is often considered in practice. In this paper, we propose a new estimation and variable selection procedure based on modal regression, where the nonparametric functions are approximated by $B$ -spline basis. The outstanding merit of the proposed variable selection procedure is that it can achieve both robustness and efficiency by introducing an additional tuning parameter (i.e., bandwidth $h$ ). Its oracle property is also established for both the parametric and nonparametric part. Moreover, we give the data-driven bandwidth selection method and propose an EM-type algorithm for the proposed method. Monte Carlo simulation study and real data example are conducted to examine the finite sample performance of the proposed method. Both the simulation results and real data analysis confirm that the newly proposed method works very well.     

Empirical likelihood for a varying coefficient partially linear model with diverging number of parameters

The purpose of this paper is two-fold. First, for the estimation or inference about the parameters of interest in semiparametric models, the commonly used plug-in estimation for infinite-dimensional nuisance parameter creates non-negligible bias, and the least favorable curve or under-smoothing is popularly employed for bias reduction in the literature. To avoid such strong structure assumptions on the models and inconvenience of estimation implementation, for the diverging number of parameters in a varying coefficient partially linear model, we adopt a bias-corrected empirical likelihood (BCEL) in this paper. This method results in the distribution of the empirical likelihood ratio to be asymptotically tractable. It can then be directly applied to construct confidence region for the parameters of interest. Second, different from all existing methods that impose strong conditions to ensure consistency of estimation when diverging the number of the parameters goes to infinity as the sample size goes to infinity, we provide techniques to show that, other than the usual regularity conditions, the consistency holds under moment conditions alone on the covariates and error with a diverging rate being even faster than those in the literature. A simulation study is carried out to assess the performance of the proposed method and to compare it with the profile least squares method. A real dataset is analyzed for illustration.     

变系数联立模型的局部线性广义矩变窗宽估计

在随机设计条件下,提出了一类变系数联立模型,运用局部线性广义矩变窗宽估计,对模型的变系数进行了估计,研究了估计量的大样本性质.利用概率论中大数定律和中心极限定理,证明了估计量的大样本性质,局部线性广义矩变窗宽估计具有相合性和渐进正态性.     

Techniques for approximating a spatially varying Euler-Bernoulli model with a constant coefficient model

Developing accurate models to describe the behaviour of a physical system often results in differential equations with spatially varying coefficients. A notable example of this that appears in many applications is the Euler-Bernoulli beam equation for transverse vibrations. This equation with spatially varying coefficients, such as when the bending stiffness or mass per unit length varies along the length of the beam, is of interest in the current research. Methods for approximating the Euler-Bernoulli equation with periodically varying coefficients have been proposed yet there is still a need for methods that approximate the more general, non-periodically varying, cases. The goal of this research is to obtain a constant coefficient Euler-Bernoulli equation that accurately approximates the original spatially varying equation using an inverse problem approach. Obtaining such an approximation has advantages in control applications where a constant coefficient model is strongly preferred for computational efficiency. The motivation for this research stems from previous work by the authors on modelling cable-harnessed structures. The spatially varying equation is solved using the Lindstedt-Poincaré perturbation method and these results are used to determine the approximate model. Multiple inverse problem methods for determining the coefficients in the approximate model are considered including metric minimization, the modal participation factor (MPF), and the proper orthogonal decomposition (POD). Continuous version of POD and MPF methods are obtained. Several wrapping patterns and boundary conditions are considered for comparison and the results are in good agreement with analytical and finite element analysis (FEA) results.     

Partially linear varying coefficient models with missing at random responses

This paper considers partially linear varying coefficient models when the response variable is missing at random. The paper uses imputation techniques to develop an omnibus specification test. The test is based on a simple modification of a Cramer von Mises functional that overcomes the curse of dimensionality often associated with the standard Cramer von Mises functional. The paper also considers estimation of the mean functional under the missing at random assumption. The proposed estimator lies in between a fully nonparametric and a parametric one and can be used, for example, to obtain a novel estimator for the average treatment effect parameter. Monte Carlo simulations show that the proposed estimator and test statistic have good finite sample properties. An empirical application illustrates the applicability of the results of the paper.     

Semiparametric variable selection for partially varying coefficient models with endogenous variables

By using instrumental variable technology and the partial group smoothly clipped absolute deviation penalty method, we propose a variable selection procedure for a class of partially varying coefficient models with endogenous variables. The proposed variable selection method can eliminate the influence of the endogenous variables. With appropriate selection of the tuning parameters, we establish the oracle property of this variable selection procedure. A simulation study is undertaken to assess the finite sample performance of the proposed variable selection procedure.     

一类高阶变系数线性非齐次微分方程的解

利用高阶变系数之间的关系,通过适当的线性变换,得到了五阶变系数线性非齐次方程常系数化的条件,给出了一类高阶变系数线性非齐次微分方程的新解法.     

Model detection and variable selection for varying coefficient models with longitudinal data

In this paper, we consider the problem of variable selection and model detection in varying coefficient models with longitudinal data. We propose a combined penalization procedure to select the significant variables, detect the true structure of the model and estimate the unknown regression coefficients simultaneously. With appropriate selection of the tuning parameters, we show that the proposed procedure is consistent in both variable selection and the separation of varying and constant coefficients, and the penalized estimators have the oracle property. Finite sample performances of the proposed method are illustrated by some simulation studies and the real data analysis.     

Empirical likelihood based inference for semiparametric varying coefficient partially linear models with error-prone linear covariates

This paper considers statistical inference for semiparametric varying coefficient partially linear models with error-prone linear covariates. An empirical likelihood based statistic for parametric component is developed to construct confidence regions. The resulting statistic is shown to be asymptotically chi-square distributed. By the empirical likelihood ratio function, the maximum empirical likelihood estimator of the parameter is defined and the asymptotic normality is shown. A simulation experiment is conducted to compare the empirical likelihood, normal based and the naive empirical likelihood methods in terms of coverage accuracies of confidence regions.     

半参数变系数部分线性模型的统计推断

本文在多种复杂数据下, 研究一类半参数变系数部分线性模型的统计推断理论和方法. 首先在纵向数据和测量误差数据等复杂数据下, 研究半参数变系数部分线性模型的经验似然推断问题, 分别提出分组的和纠偏的经验似然方法. 该方法可以有效地处理纵向数据的组内相关性给构造经验似然比函数所带来的困难. 其次在测量误差数据和缺失数据等复杂数据下, 研究模型的变量选择问题, 分别提出一个“纠偏” 的和基于借补值的变量选择方法. 该变量选择方法可以同时选择参数分量及非参数分量中的重要变量, 并且变量选择与回归系数的估计同时进行. 通过选择适当的惩罚参数, 证明该变量选择方法可以相合地识别出真实模型, 并且所得的正则估计具有oracle 性质.     

Influence diagnostics in the varying coefficient model with longitudinal data

One or few observations can be highly influential on estimates of regression coefficients in the linear regression model. In this paper we derive influence diagnostics for the varying coefficients model with longitudinal data. We note that diagnostics in this context is quite different from the classical regression model in the sense that regression coefficients vary as time varies. A version of Cook’s distance is suggested to reflect this specific aspect of varying coefficient model. An algorithm to present some guidelines to determine influential observations deserving special attention is developed. An illustrative example based on the AIDS data is also given.     

Variable selection for semiparametric varying coefficient partially linear errors-in-variables models

This paper focuses on the variable selections for semiparametric varying coefficient partially linear models when the covariates in the parametric and nonparametric components are all measured with errors. A bias-corrected variable selection procedure is proposed by combining basis function approximations with shrinkage estimations. With appropriate selection of the tuning parameters, the consistency of the variable selection procedure and the oracle property of the regularized estimators are established. A simulation study and a real data application are undertaken to evaluate the finite sample performance of the proposed method.     

Spline estimator for ultra-high dimensional partially linear varying coefficient models

Annals of the Institute of Statistical Mathematics - In this paper, we simultaneously study variable selection and estimation problems for sparse ultra-high dimensional partially linear varying...     

空间半参数变系数部分线性回归中的分位数估计

本文研究了空间数据变系数部分线性回归中的分位数估计. 模型中的参数估计量通过未知系数函数的分段多项式逼近得到, 而未知系数函数的估计量通过将参数估计量代入模型中并通过局部线性逼近得到. 文中推导了未知参数向量估计量的渐近分布, 并建立了未知系数函数估计量在内点及边界点的渐近分布. 通过Monte Carlo 模拟研究了估计量的有限样本性质.     

Bias-corrected statistical inference for partially linear varying coefficient errors-in-variables models with restricted condition

In this paper, we consider the statistical inference for the partially liner varying coefficient model with measurement error in the nonparametric part when some prior information about the parametric part is available. The prior information is expressed in the form of exact linear restrictions. Two types of local bias-corrected restricted profile least squares estimators of the parametric component and nonparametric component are conducted, and their asymptotic properties are also studied under some regularity conditions. Moreover, we compare the efficiency of the two kinds of parameter estimators under the criterion of Lo?ner ordering. Finally, we develop a linear hypothesis test for the parametric component. Some simulation studies are conducted to examine the finite sample performance for the proposed method. A real dataset is analyzed for illustration.     

Numerical study of a diffusive epidemic model of influenza with variable transmission coefficient

A diffusive epidemic model for H1N1 influenza is formulated with a view to gain basic understanding of the virus behavior. All newborns are assumed to be susceptible. Mortality rate for infective individuals in the population is assumed to be greater than natural mortality rate. Latent, infectious and immune periods are assumed to be constants throughout this study. The numerical solutions of this model are carried out under three different initial populations distribution. In order to investigate the effect of the disease transmission coefficient on the spread of disease, β is taken to be constant as well as a function of seasonally varying time t and a function of spatial variable x  . The threshold quantity (_R0)$({\mathfrak{R}}_0)$ that governs the disease dynamics is derived. Numerical simulation shows that the system supports the existence of sustained and damped oscillations depending on initial populations distribution, the disease transmission rate and diffusion.     

Variable selection for generalized varying coefficient partially linear models with diverging number of parameters

Semiparametric models with diverging number of predictors arise in many contemporary scientific areas.Variable selection for these models consists of two components:model selection for non-parametric components and selection of significant variables for the parametric portion.In this paper,we consider a variable selection procedure by combining basis function approximation with SCAD penalty.The proposed procedure simultaneously selects significant variables in the parametric components and the nonparametric components.With appropriate selection of tuning parameters,we establish the consistency and sparseness of this procedure.     

On estimation and inference in a partially linear hazard model with varying coefficients

We study estimation and inference in a marginal proportional hazards model that can handle (1) linear effects, (2) non-linear effects and (3) interactions between covariates. The model under consideration is an amalgamation of three existing marginal proportional hazards models studied in the literature. Developing an estimation and inference procedure with desirable properties for the amalgamated model is rather challenging due to the co-existence of all three effects listed above. Much of the existing literature has avoided the problem by considering narrow versions of the model. The object of this paper is to show that an estimation and inference procedure that accommodates all three effects is within reach. We present a profile partial-likelihood approach for estimating the unknowns in the amalgamated model with the resultant estimators of the unknown parameters being root- \(n\) consistent and the estimated functions achieving optimal convergence rates. Asymptotic normality is also established for the estimators.