Variable selection for semiparametric varying coefficient partially linear model based on modal regression with missing data

Abstract

In this article, we focus on the variable selection for semiparametric varying coefficient partially linear model with response missing at random. Variable selection is proposed based on modal regression, where the non parametric functions are approximated by B-spline basis. The proposed procedure uses SCAD penalty to realize variable selection of parametric and nonparametric components simultaneously. Furthermore, we establish the consistency, the sparse property and asymptotic normality of the resulting estimators. The penalty estimation parameters value of the proposed method is calculated by EM algorithm. Simulation studies are carried out to assess the finite sample performance of the proposed variable selection procedure.     

Variable selection in finite mixture of semi-parametric regression models

Abstract

In this paper we are concerned with variable selection in finite mixture of semiparametric regression models. This task consists of model selection for non parametric component and variable selection for parametric part. Thus, we encountered separate model selections for every non parametric component of each sub model. To overcome this computational burden, we introduced a class of variable selection procedures for finite mixture of semiparametric regression models using penalized approach for variable selection. It is shown that the new method is consistent for variable selection. Simulations show that the performance of proposed method is good, and it consequently improves pervious works in this area and also requires much less computing power than existing methods.     

M-estimation and model identification based on double SCAD penalization

M-estimation is a widely used method for robust statistical inference. In this article, using a B-spline series approximation with a double smoothly clipped absolute deviation penalization, we solve the problem of simultaneous variable selection and parametric component identification in a non parametric additive model. The theoretical properties of the double non concave penalized M-estimation are established. The proposed approach is resistant to heavy-tailed errors or outliers in the responses. Simulation studies for finite-sample cases are conducted and a real dataset is also analyzed for illustration of this new approach.     

WLAD-LASSO method for robust estimation and variable selection in partially linear models

This paper focuses on robust estimation and variable selection for partially linear models. We combine the weighted least absolute deviation (WLAD) regression with the adaptive least absolute shrinkage and selection operator (LASSO) to achieve simultaneous robust estimation and variable selection for partially linear models. Compared with the LAD-LASSO method, the WLAD-LASSO method will resist to the heavy-tailed errors and outliers in the parametric components. In addition, we estimate the unknown smooth function by a robust local linear regression. Under some regular conditions, the theoretical properties of the proposed estimators are established. We further examine finite-sample performance of the proposed procedure by simulation studies and a real data example.     

Semiparametric statistical inferences for longitudinal data with nonparametric covariance modelling

In this paper, semiparametric modelling for longitudinal data with an unstructured error process is considered. We propose a partially linear additive regression model for longitudinal data in which within-subject variances and covariances of the error process are described by unknown univariate and bivariate functions, respectively. We provide an estimating approach in which polynomial splines are used to approximate the additive nonparametric components and the within-subject variance and covariance functions are estimated nonparametrically. Both the asymptotic normality of the resulting parametric component estimators and optimal convergence rate of the resulting nonparametric component estimators are established. In addition, we develop a variable selection procedure to identify significant parametric and nonparametric components simultaneously. We show that the proposed SCAD penalty-based estimators of non-zero components have an oracle property. Some simulation studies are conducted to examine the finite-sample performance of the proposed estimation and variable selection procedures. A real data set is also analysed to demonstrate the usefulness of the proposed method.     

Simultaneous structure estimation and variable selection in partial linear varying coefficient models for longitudinal data

Partial linear varying coefficient models (PLVCM) are often considered for analysing longitudinal data for a good balance between flexibility and parsimony. The existing estimation and variable selection methods for this model are mainly built upon which subset of variables have linear or varying effect on the response is known in advance, or say, model structure is determined. However, in application, this is unreasonable. In this work, we propose a simultaneous structure estimation and variable selection method, which can do simultaneous coefficient estimation and three types of selections: varying and constant effects selection, relevant variable selection. It can be easily implemented in one step by employing a penalized M-type regression, which uses a general loss function to treat mean, median, quantile and robust mean regressions in a unified framework. Consistency in the three types of selections and oracle property in estimation are established as well. Simulation studies and real data analysis also confirm our method.     

Estimation and variable selection for partially functional linear models

In this paper, a new estimation procedure based on composite quantile regression and functional principal component analysis (PCA) method is proposed for the partially functional linear regression models (PFLRMs). The proposed estimation method can simultaneously estimate both the parametric regression coefficients and functional coefficient components without specification of the error distributions. The proposed estimation method is shown to be more efficient empirically for non-normal random error, especially for Cauchy error, and almost as efficient for normal random errors. Furthermore, based on the proposed estimation procedure, we use the penalized composite quantile regression method to study variable selection for parametric part in the PFLRMs. Under certain regularity conditions, consistency, asymptotic normality, and Oracle property of the resulting estimators are derived. Simulation studies and a real data analysis are conducted to assess the finite sample performance of the proposed methods.     

Efficiently weighted estimating equations with application to proportional excess hazards

A general approach to estimation, that can lead to efficient estimation in two stages, is presented. The method will not always be available, but sufficient conditions for efficiency are provided together with four examples of its use: (1) estimation of the odds ratio in 1:M matched case-control studies with a dichotomous exposure variable; (2) estimation of the relative hazard in a two-sample survival setting; (3) estimation of the regression parameters in the proportional excess hazards model; and (4) estimation in a partly linear parametric additive hazards model. The method depends upon finding a family of weighted estimating equations, which includes a simple initial equation yielding a consistent estimate and also an equation that yields an efficient estimate, provided the optiomal weights are used.     

A new model selection procedure for finite mixture regression models

Abstract

In this article, we propose a new penalized-likelihood method to conduct model selection for finite mixture of regression models. The penalties are imposed on mixing proportions and regression coefficients, and hence order selection of the mixture and the variable selection in each component can be simultaneously conducted. The consistency of order selection and the consistency of variable selection are investigated. A modified EM algorithm is proposed to maximize the penalized log-likelihood function. Numerical simulations are conducted to demonstrate the finite sample performance of the estimation procedure. The proposed methodology is further illustrated via real data analysis.     

A novel regularization method for estimation and variable selection in multi-index models

Multi-index models have attracted much attention recently as an approach to circumvent the curse of dimensionality when modeling high-dimensional data. This paper proposes a novel regularization method, called MAVE-glasso, for simultaneous parameter estimation and variable selection in multi-index models. The advantages of the proposed method include transformation invariance, automatic variable selection, automatic removal of noninformative observations, and row-wise shrinkage. An efficient row-wise coordinate descent algorithm is proposed to calculate the estimates. Simulation and real examples are used to demonstrate the excellent performance of MAVE-glasso.     

Quantile regression for robust estimation and variable selection in partially linear varying-coefficient models

In this paper, we develop a new estimation procedure based on quantile regression for semiparametric partially linear varying-coefficient models. The proposed estimation approach is empirically shown to be much more efficient than the popular least squares estimation method for non-normal error distributions, and almost not lose any efficiency for normal errors. Asymptotic normalities of the proposed estimators for both the parametric and nonparametric parts are established. To achieve sparsity when there exist irrelevant variables in the model, two variable selection procedures based on adaptive penalty are developed to select important parametric covariates as well as significant nonparametric functions. Moreover, both these two variable selection procedures are demonstrated to enjoy the oracle property under some regularity conditions. Some Monte Carlo simulations are conducted to assess the finite sample performance of the proposed estimators, and a real-data example is used to illustrate the application of the proposed methods.     

A Semiparametric Regression Model for Longitudinal Data with Non‐stationary Errors

Motivated by the need to analyze the National Longitudinal Surveys data, we propose a new semiparametric longitudinal mean‐covariance model in which the effects on dependent variable of some explanatory variables are linear and others are non‐linear, while the within‐subject correlations are modelled by a non‐stationary autoregressive error structure. We develop an estimation machinery based on least squares technique by approximating non‐parametric functions via B‐spline expansions and establish the asymptotic normality of parametric estimators as well as the rate of convergence for the non‐parametric estimators. We further advocate a new model selection strategy in the varying‐coefficient model framework, for distinguishing whether a component is significant and subsequently whether it is linear or non‐linear. Besides, the proposed method can also be employed for identifying the true order of lagged terms consistently. Monte Carlo studies are conducted to examine the finite sample performance of our approach, and an application of real data is also illustrated.     

Bayesian bridge quantile regression

Regularization methods for simultaneous variable selection and coefficient estimation have been shown to be effective in quantile regression in improving the prediction accuracy. In this article, we propose the Bayesian bridge for variable selection and coefficient estimation in quantile regression. A simple and efficient Gibbs sampling algorithm was developed for posterior inference using a scale mixture of uniform representation of the Bayesian bridge prior. This is the first work to discuss regularized quantile regression with the bridge penalty. Both simulated and real data examples show that the proposed method often outperforms quantile regression without regularization, lasso quantile regression, and Bayesian lasso quantile regression.     

Model Selection,Transformations and Variance Estimation in Nonlinear Regression

The results of analyzing experimental data using a parametric model may heavily depend on the chosen model for regression and variance functions, moreover also on a possibly underlying preliminary transformation of the variables. In this paper we propose and discuss a complex procedure which consists in a simultaneous selection of parametric regression and variance models from a relatively rich model class and of Box-Cox variable transformations by minimization of a cross-validation criterion. For this it is essential to introduce modifications of the standard cross-validation criterion adapted to each of the following objectives: 1. estimation of the unknown regression function, 2. prediction of future values of the response variable, 3. calibration or 4. estimation of some parameter with a certain meaning in the corresponding field of application. Our idea of a criterion oriented combination of procedures (which usually if applied, then in an independent or sequential way) is expected to lead to more accurate results. We show how the accuracy of the parameter estimators can be assessed by a “moment oriented bootstrap procedure", which is an essential modification of the “wild bootstrap” of Härdle and Mammen by use of more accurate variance estimates. This new procedure and its refinement by a bootstrap based pivot (“double bootstrap”) is also used for the construction of confidence, prediction and calibration intervals. Programs written in Splus which realize our strategy for nonlinear regression modelling and parameter estimation are described as well. The performance of the selected model is discussed, and the behaviour of the procedures is illustrated, e.g., by an application in radioimmunological assay.     

Statistical properties of parametric estimators for Markov chain vectors based on copula models

To estimate and measure risks, two key classes of dependence relationship must be identified: temporal dependence and contemporaneous dependence. In this paper, we propose a parametric estimation model that uses a three-stage pseudo maximum likelihood estimation (3SPMLE), and we investigate the consistency and asymptotic normality of parametric estimators. The proposed model combines the concept of a copula and the methods of parametric estimators of two-stage pseudo maximum likelihood estimation (2SPMLE). The selection of a copula model that best captures the dependence structure is a critical problem. To solve this problem, we propose a model selection method that is based on the parametric pseudo-likelihood ratio under the 3SPMLE for stationary Markov vector-type models.     

Penalized least-squares estimation for regression coefficients in high-dimensional partially linear models

We consider a partially linear model with diverging number of groups of parameters in the parametric component. The variable selection and estimation of regression coefficients are achieved simultaneously by using the suitable penalty function for covariates in the parametric component. An MM-type algorithm for estimating parameters without inverting a high-dimensional matrix is proposed. The consistency and sparsity of penalized least-squares estimators of regression coefficients are discussed under the setting of some nonzero regression coefficients with very small values. It is found that the root p_n/n-consistency and sparsity of the penalized least-squares estimators of regression coefficients cannot be given consideration simultaneously when the number of nonzero regression coefficients with very small values is unknown, where p_n and n, respectively, denote the number of regression coefficients and sample size. The finite sample behaviors of penalized least-squares estimators of regression coefficients and the performance of the proposed algorithm are studied by simulation studies and a real data example.     

面板数据的自适应Lasso分位回归方法研究

如何在对参数进行估计的同时自动选择重要解释变量,一直是面板数据分位回归模型中讨论的热点问题之一。通过构造一种含多重随机效应的贝叶斯分层分位回归模型,在假定固定效应系数先验服从一种新的条件Laplace分布的基础上,给出了模型参数估计的Gibbs抽样算法。考虑到不同重要程度的解释变量权重系数压缩程度应该不同,所构造的先验信息具有自适应性的特点,能够准确地对模型中重要解释变量进行自动选取,且设计的切片Gibbs抽样算法能够快速有效地解决模型中各个参数的后验均值估计问题。模拟结果显示,新方法在参数估计精确度和变量选择准确度上均优于现有文献的常用方法。通过对中国各地区多个宏观经济指标的面板数据进行建模分析,演示了新方法估计参数与挑选变量的能力。     

Component Selection in the Additive Regression Model

Abstract. Similar to variable selection in the linear model, selecting significant components in the additive model is of great interest. However, such components are unknown, unobservable functions of independent variables. Some approximation is needed. We suggest a combination of penalized regression spline approximation and group variable selection, called the group‐bridge‐type spline method (GBSM), to handle this component selection problem with a diverging number of correlated variables in each group. The proposed method can select significant components and estimate non‐parametric additive function components simultaneously. To make the GBSM stable in computation and adaptive to the level of smoothness of the component functions, weighted power spline bases and projected weighted power spline bases are proposed. Their performance is examined by simulation studies. The proposed method is extended to a partial linear regression model analysis with real data, and gives reliable results.     

NEW EFFICIENT ESTIMATION AND VARIABLE SELECTION METHODS FOR SEMIPARAMETRIC VARYING-COEFFICIENT PARTIALLY LINEAR MODELS

The complexity of semiparametric models poses new challenges to statistical inference and model selection that frequently arise from real applications. In this work, we propose new estimation and variable selection procedures for the semiparametric varying-coefficient partially linear model. We first study quantile regression estimates for the nonparametric varying-coefficient functions and the parametric regression coefficients. To achieve nice efficiency properties, we further develop a semiparametric composite quantile regression procedure. We establish the asymptotic normality of proposed estimators for both the parametric and nonparametric parts and show that the estimators achieve the best convergence rate. Moreover, we show that the proposed method is much more efficient than the least-squares-based method for many non-normal errors and that it only loses a small amount of efficiency for normal errors. In addition, it is shown that the loss in efficiency is at most 11.1% for estimating varying coefficient functions and is no greater than 13.6% for estimating parametric components. To achieve sparsity with high-dimensional covariates, we propose adaptive penalization methods for variable selection in the semiparametric varying-coefficient partially linear model and prove that the methods possess the oracle property. Extensive Monte Carlo simulation studies are conducted to examine the finite-sample performance of the proposed procedures. Finally, we apply the new methods to analyze the plasma beta-carotene level data.     

Functional Partial Linear Single‐index Model

This paper deals with the problem of predicting the real‐valued response variable using explanatory variables containing both multivariate random variable and random curve. The proposed functional partial linear single‐index model treats the multivariate random variable as linear part and the random curve as functional single‐index part, respectively. To estimate the non‐parametric link function, the functional single‐index and the parameters in the linear part, a two‐stage estimation procedure is proposed. Compared with existing semi‐parametric methods, the proposed approach requires no initial estimation and iteration. Asymptotical properties are established for both the parameters in the linear part and the functional single‐index. The convergence rate for the non‐parametric link function is also given. In addition, asymptotical normality of the error variance is obtained that facilitates the construction of confidence region and hypothesis testing for the unknown parameter. Numerical experiments including simulation studies and a real‐data analysis are conducted to evaluate the empirical performance of the proposed method.