Variable Selection for Semiparametric Partially Linear Covariate-Adjusted Regression Models

In this article, the partially linear covariate-adjusted regression models are considered, and the penalized least-squares procedure is proposed to simultaneously select variables and estimate the parametric components. The rate of convergence and the asymptotic normality of the resulting estimators are established under some regularization conditions. With the proper choices of the penalty functions and tuning parameters, it is shown that the proposed procedure can be as efficient as the oracle estimators. Some Monte Carlo simulation studies and a real data application are carried out to assess the finite sample performances for the proposed method.     

Smooth-Threshold GEE Variable Selection in High-Dimensional Partially Linear Models with Longitudinal Data

We consider the problem of variable selection in high-dimensional partially linear models with longitudinal data. A variable selection procedure is proposed based on the smooth-threshold generalized estimating equation (SGEE). The proposed procedure automatically eliminates inactive predictors by setting the corresponding parameters to be zero, and simultaneously estimates the nonzero regression coefficients by solving the SGEE. We establish the asymptotic properties in a high-dimensional framework where the number of covariates p_n increases as the number of clusters n increases. Extensive Monte Carlo simulation studies are conducted to examine the finite sample performance of the proposed variable selection procedure.     

Bridge Estimators in the Partially Linear Model with High Dimensionality

This article studies variable selection and parameter estimation in the partially linear model when the number of covariates in the linear part increases to infinity. Using the bridge penalty method, we succeed in selecting the important covariates of the linear part. Under regularity conditions, we have shown that the bridge penalized estimator of the parametric part enjoys the oracle property. We also obtain the convergence rate of the estimator of the nonparametric part. Simulation studies show that the bridge estimator performs as well as the oracle estimator for the partially linear model. An application is analyzed to illustrate the bridge procedure.     

Variable Selection in Semiparametric Quantile Modeling for Longitudinal Data

We propose a penalized quantile regression for partially linear varying coefficient (VC) model with longitudinal data to select relevant non parametric and parametric components simultaneously. Selection consistency and oracle property are established. Furthermore, if linear part and VC part are unknown, we propose a new unified method, which can do three types of selections: separation of varying and constant effects, selection of relevant variables, and it can be carried out conveniently in one step. 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.     

Asymptotic Normality for the Partially Linear EV Models with Longitudinal Data

In this article, we consider a partially linear EV regression model under longitudinal data. By using a weighted kernel method and modified least-squared method, the estimators of unknown parameter, the unknown function are constructed and the asymptotic normality of the estimators are derived. Simulation studies are conducted to illustrate the finite-sample performance of the proposed method.     

Robust Variable Selection in Linear Mixed Models

In this article, we develop a robust variable selection procedure jointly for fixed and random effects in linear mixed models for longitudinal data. We propose a penalized robust estimator for both the regression coefficients and the variance of random effects based on a re-parametrization of the linear mixed models. Under some regularity conditions, we show the oracle properties of the proposed robust variable selection method. Simulation study shows the robustness of the proposed method against outliers. In the end, the proposed methods is illustrated in the analysis of a real data set.     

Local Linear Regression in Proportional Hazards Model with Censored Data

In this article we study the method of nonparametric regression based on a transformation model, under which an unknown transformation of the survival time is nonlinearly, even more, nonparametrically, related to the covariates with various error distributions, which are parametrically specified with unknown parameters. Local linear approximations and locally weighted least squares are applied to obtain estimators for the effects of covariates with censored observations. We show that the estimators are consistent and asymptotically normal. This transformation model, coupled with local linear approximation techniques, provides many alternatives to the more general proportional hazards models with nonparametric covariates.     

Adaptive Lasso Variable Selection for the Accelerated Failure Models

This article considers the adaptive lasso procedure for the accelerated failure time model with multiple covariates based on weighted least squares method, which uses Kaplan-Meier weights to account for censoring. The adaptive lasso method can complete the variable selection and model estimation simultaneously. Under some mild conditions, the estimator is shown to have sparse and oracle properties. We use Bayesian Information Criterion (BIC) for tuning parameter selection, and a bootstrap variance approach for standard error. Simulation studies and two real data examples are carried out to investigate the performance of the proposed method.     

Empirical Likelihood for Censored Linear Regression and Variable Selection

The linear regression model for right censored data, also known as the accelerated failure time model using the logarithm of survival time as the response variable, is a useful alternative to the Cox proportional hazards model. Empirical likelihood as a non‐parametric approach has been demonstrated to have many desirable merits thanks to its robustness against model misspecification. However, the linear regression model with right censored data cannot directly benefit from the empirical likelihood for inferences mainly because of dependent elements in estimating equations of the conventional approach. In this paper, we propose an empirical likelihood approach with a new estimating equation for linear regression with right censored data. A nested coordinate algorithm with majorization is used for solving the optimization problems with non‐differentiable objective function. We show that the Wilks' theorem holds for the new empirical likelihood. We also consider the variable selection problem with empirical likelihood when the number of predictors can be large. Because the new estimating equation is non‐differentiable, a quadratic approximation is applied to study the asymptotic properties of penalized empirical likelihood. We prove the oracle properties and evaluate the properties with simulated data. We apply our method to a Surveillance, Epidemiology, and End Results small intestine cancer dataset.     

Smoothing Spline Estimation for Partially Linear Single-index Models

In this article, the partially linear single-index models are discussed based on smoothing spline and average derivative estimation method. This proposed technique consists of two stages: one is to estimate the vector parameter in the linear part using the smoothing cubic spline method, simultaneously, obtaining the estimator of unknown single-index function; the other is to estimate the single-index coefficients in the single-index part by the using average derivative estimator procedure. Some simulated and real examples are presented to illustrate the performance of this method.     

两个部分线性模型的比较

内容提要:对于两个部分线性模型参数部分中模型系数是否相等的检验问题,本文基于比较原假设与备择假设下模型拟合的残差平方和的思想构造了检验统计量,并给出了计算检验_^p* 值的F分布逼近法。     

Joint Mean-Covariance Models with Applications to Longitudinal Data in Partially Linear Model

Semiparametric regression models and estimating covariance functions are very useful in longitudinal study. Unfortunately, challenges arise in estimating the covariance function of longitudinal data collected at irregular time points. In this article, for mean term, a partially linear model is introduced and for covariance structure, a modified Cholesky decomposition approach is proposed to heed the positive-definiteness constraint. We estimate the regression function by using the local linear technique and propose quasi-likelihood estimating equations for both the mean and covariance structures. Moreover, asymptotic normality of the resulting estimators is established. Finally, simulation study and real data analysis are used to illustrate the proposed approach.     

Estimation for Location-Scale Models with Censored Data

This article considers a class of estimators for the location and scale parameters in the location-scale model based on ‘synthetic data’ when the observations are randomly censored on the right. The asymptotic normality of the estimators is established using counting process and martingale techniques when the censoring distribution is known and unknown, respectively. In the case when the censoring distribution is known, we show that the asymptotic variances of this class of estimators depend on the data transformation and have a lower bound which is not achievable by this class of estimators. However, in the case that the censoring distribution is unknown and estimated by the Kaplan–Meier estimator, this class of estimators has the same asymptotic variance and attains the lower bound for variance for the case of known censoring distribution. This is different from censored regression analysis, where asymptotic variances depend on the data transformation. Our method has three valuable advantages over the method of maximum likelihood estimation. First, our estimators are available in a closed form and do not require an iterative algorithm. Second, simulation studies show that our estimators being moment-based are comparable to maximum likelihood estimators and outperform them when sample size is small and censoring rate is high. Third, our estimators are more robust to model misspecification than maximum likelihood estimators. Therefore, our method can serve as a competitive alternative to the method of maximum likelihood in estimation for location-scale models with censored data. A numerical example is presented to illustrate the proposed method.     

联合广义线性模型中的变量选择

在联合广义线性模型中,散度参数与均值都被赋予了广义线性模型的结构,本文主要考虑在只有分布的一阶矩和二阶矩指定的条件下,联合广义线性模型中均值部分的变量选择问题。本文采用广义拟似然函数,提出了新的模型选择准则(EAIC);该准则是Akaike信息准则的推广。论文通过模拟研究验证了该准则的效果。     

Covariate-Adjusted Partially Linear Regression Models

Motivated by covariate-adjusted regression (CAR) proposed by Sentürk and Müller (2005 Sentürk , D. , Müller , H. G. ( 2005 ). Covariate-adjusted regression . Biometrika 92 : 75 – 89 .[Crossref], [Web of Science ®] , [Google Scholar]) and an application problem, in this article we introduce and investigate a covariate-adjusted partially linear regression model (CAPLM), in which both response and predictor vector can only be observed after being distorted by some multiplicative factors, and an additional variable such as age or period is taken into account. Although our model seems to be a special case of covariate-adjusted varying coefficient model (CAVCM) given by Sentürk (2006 Sentürk , D. ( 2006 ). Covariate-adjusted varying coefficient models . Biostatistics 7 : 235 – 251 .[Crossref], [PubMed], [Web of Science ®] , [Google Scholar]), the data types of CAPLM and CAVCM are basically different and then the methods for inferring the two models are different. In this article, the estimate method motivated by Cui et al. (2008 Cui , X. , Guo , W. S. , Lin , L. , Zhu , L. X. ( 2008 ). Covariate-adjusted nonlinear regression . Ann. Statist. 37 : 1839 – 1870 . [Google Scholar]) is employed to infer the new model. Furthermore, under some mild conditions, the asymptotic normality of estimator for the parametric component is obtained. Combined with the consistent estimate of asymptotic covariance, we obtain confidence intervals for the regression coefficients. Also, some simulations and a real data analysis are made to illustrate the new model and methods.     

A Robust Variable Selection to t-type Joint Generalized Linear Models via Penalized t-type Pseudo-likelihood

Although the t-type estimator is a kind of M-estimator with scale optimization, it has some advantages over the M-estimator. In this article, we first propose a t-type joint generalized linear model as a robust extension to the classical joint generalized linear models for modeling data containing extreme or outlying observations. Next, we develop a t-type pseudo-likelihood (TPL) approach, which can be viewed as a robust version to the existing pseudo-likelihood (PL) approach. To determine which variables significantly affect the variance of the response variable, we then propose a unified penalized maximum TPL method to simultaneously select significant variables for the mean and dispersion models in t-type joint generalized linear models. Thus, the proposed variable selection method can simultaneously perform parameter estimation and variable selection in the mean and dispersion models. With appropriate selection of the tuning parameters, we establish the consistency and the oracle property of the regularized estimators. Simulation studies are conducted to illustrate the proposed methods.     

Variable Selection for Semiparametric Varying Coefficient Partially Linear Errors-in-Variables (EV) Model with Missing Response

This paper focuses on the variable selection for semiparametric varying coefficient partially linear model when the covariates are measured with additive errors and the response is missing. An adaptive lasso estimator and the smoothly clipped absolute deviation estimator as a comparison for the parameters are proposed. With the proper selection of regularization parameter, the sampling properties including the consistency of the two procedures and the oracle properties are established. Furthermore, the algorithms and corresponding standard error formulas are discussed. A simulation study is carried out to assess the finite sample performance of the proposed methods.     

Hierarchical-Likelihood Approach for Mixed Linear Models with Censored Data

Mixed linear models describe the dependence via random effects in multivariate normal survival data. Recently they have received considerable attention in the biomedical literature. They model the conditional survival times, whereas the alternative frailty model uses the conditional hazard rate. We develop an inferential method for the mixed linear model via Lee and Nelder's (1996) hierarchical-likelihood (h-likelihood). Simulation and a practical example are presented to illustrate the new method.     

Restricted Estimation in Partially Linear Errors-in-Variables Models

As a compromise between parametric regression and nonparametric regression, partially linear models are frequently used in statistical modelling. This article considers statistical inference for this semiparametric model when the linear covariate is measured with additive error and some additional linear restrictions on the parametric component are assumed to hold. We propose a restricted corrected profile least-squares estimator for the parametric component, and study the asymptotic normality of the estimator. To test hypothesis on the parametric component, we construct a Wald test statistic and obtain its limiting distribution. Some simulation studies are conducted to illustrate our approaches.     

Empirical-likelihood-based Test for Partially Linear Single-index Models with Error-prone Linear Covariates

In this article, we consider whether the empirical likelihood ratio (ELR) test is applicable to testing for serial correlation in the partially linear single-index models (PLSIM) with error-prone linear covariates. It is shown that under the null hypothesis the proposed ELR statistic follows asymptotically a χ²-distribution with the scale constant and the degrees of freedom. A comparison between the ELR and the normal approximation method is also considered. Both simulated and real data examples are used to illustrate our proposed methodology.