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.     

Variable Selection for Partially Linear Models with Randomly Censored Data

This article proposes a variable selection procedure for partially linear models with right-censored data via penalized least squares. We apply the SCAD penalty to select significant variables and estimate unknown parameters simultaneously. The sampling properties for the proposed procedure are investigated. The rate of convergence and the asymptotic normality of the proposed estimators are established. Furthermore, the SCAD-penalized estimators of the nonzero coefficients are shown to have the asymptotic oracle property. In addition, an iterative algorithm is proposed to find the solution of the penalized least squares. Simulation studies are conducted to examine the finite sample performance of the proposed method.     

On the Preliminary Test Backfitting and Speckman Estimators in Partially Linear Models and Numerical Comparisons

Przystalski and Krajewski (2007 Przystalski , M. , Krajewski , P. ( 2007 ). Constrained estimators of treatment parameters in semiparametric models . Statist. Probab. Lett. 77 : 914 – 919 .[Crossref], [Web of Science ®] , [Google Scholar]) proposed the restricted backfitting (RBCF) estimator and restricted Speckman (RSPC) estimator for the treatment effects in a partially linear model when some additional exact linear restrictions are assumed to hold. In this article, we introduce the preliminary test backfitting (PTBCF) estimator and preliminary test Speckman (PTSPC) estimator when the validity of the restrictions is suspected. Performances of the proposed estimators are examined with respect to the mean squared error (MSE) criterion. In addition, numerical behaviors of the proposed estimators are illustrated and compared via a Monte Carlo simulation study.     

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.     

Jackknife Estimation for Smooth Functions of the Parametric Component in Partially Linear Regression Models

Abstract

It is known that due to the existence of the nonparametric component, the usual estimators for the parametric component or its function in partially linear regression models are biased. Sometimes this bias is severe. To reduce the bias, we propose two jackknife estimators and compare them with the naive estimator. All three estimators are shown to be asymptotically equivalent and asymptotically normally distributed under some regularity conditions. However, through simulation we demonstrate that the jackknife estimators perform better than the naive estimator in terms of bias when the sample size is small to moderate. To make our results more useful, we also construct consistent estimators of the asymptotic variance, which are robust against heterogeneity of the error variances.     

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.     

Penalized Empirical Likelihood via Bridge Estimator in Cox's Proportional Hazard Model

We propose the penalized empirical likelihood method via bridge estimator in Cox's proportional hazard model for parameter estimation and variable selection. Under reasonable conditions, we show that penalized empirical likelihood in Cox's proportional hazard model has oracle property. A penalized empirical likelihood ratio for the vector of regression coefficients is defined and its limiting distribution is a chi-square distributions. The advantage of penalized empirical likelihood as a nonparametric likelihood approach is illustrated in testing hypothesis and constructing confidence sets. The method is illustrated by extensive simulation studies and a real example.     

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.     

Profile Likelihood Inferences on the Partially Linear Model with a Diverging Number of Parameters

In this article, we study the profile likelihood estimation and inference on the partially linear model with a diverging number of parameters. Polynomial splines are applied to estimate the nonparametric component and we focus on constructing profile likelihood ratio statistic to examine the testing problem for the parametric component in the partially linear model. Under some regularity conditions, the asymptotic distribution of profile likelihood ratio statistic is proposed when the number of parameters grows with the sample size. Numerical studies confirm our theory.     

Empirical Likelihood for Partially Linear Single-Index Errors-in-Variables Model

In this article, we consider the application of the empirical likelihood method to a partially linear single-index model. We focus on the case where some covariates are measured with additive errors. It is shown that the empirical log-likelihood ratio at the true parameter converges to the standard chi-square distribution. Simulations show that the proposed confidence region has coverage probability which is closer to the nominal level, as well as narrower than those of normal approximation method. A real data example is given.     

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.     

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.     

Risk Comparison of Inequality Constrained Estimators in the Heteroscedastic Linear Model

There exist many studies which treat the inequality and/or interval constraints on coefficients in the homoscedastic linear regression model. However, the sampling performance of the inequality constrained estimators in the heteroscedastic linear model has not been examined. This paper considers the inequality constrained estimators in the heteroscedastic linear regression model and derives their risks under a quadratic loss function. Furthermore, using the inequality constrained estimators, we introduce a pre-test estimator which might be employed after the test for homoscedasticity and derive its risk. In addition, the risk performance of these estimators is evaluated numerically.     

Adjustive Liu-Type Estimators in Linear Regression Models

In this article, we aim to put forward the notion of adjustive Liu-type estimator (ALTE) in the linear regression model. First, the explicit expression of the optimal selection of the adjustive factors is derived under the PRESS criterion through matrix techniques. Then, the results are applied to the dataset on Portland cement. Moreover, to select biasing parameters from the theoretical point of view, we extend ALTE to the generalized version (GALTE) and obtained the optimal ones. The results of the Portland cement data show that ALTE's and GALTE's can substantially improve the ordinary least squares estimator and Liu-type estimators.     

Estimation in Partially Linear Single-Index Models with Missing Covariates

In this article, we consider a partially linear single-index model Y = g(Z ^τθ₀) + X ^τβ₀ + ? when the covariate X may be missing at random. We propose weighted estimators for the unknown parametric and nonparametric part by applying weighted estimating equations. We establish normality of the estimators of the parameters and asymptotic expansion for the estimator of the nonparametric part when the selection probabilities are unknown. Simulation studies are also conducted to illustrate the finite sample properties of these estimators.     

On the Stochastic Restricted Almost Unbiased Estimators in Linear Regression Model

In this article, the stochastic restricted almost unbiased ridge regression estimator and stochastic restricted almost unbiased Liu estimator are proposed to overcome the well-known multicollinearity problem in linear regression model. The quadratic bias and mean square error matrix of the proposed estimators are derived and compared. Furthermore, a numerical example and a Monte Carlo simulation are given to illustrate some of the theoretical results.     

A Partially Linear Model Using a Gaussian Process Prior

A partially linear model is a semiparametric regression model that consists of parametric and nonparametric regression components in an additive form. In this article, we propose a partially linear model using a Gaussian process regression approach and consider statistical inference of the proposed model. Based on the proposed model, the estimation procedure is described by posterior distributions of the unknown parameters and model comparisons between parametric representation and semi- and nonparametric representation are explored. Empirical analysis of the proposed model is performed with synthetic data and real data applications.     

The Superiorities of Bayes Linear Minimum Risk Estimation in Linear Model

In this article, the Bayes linear minimum risk estimator (BLMRE) of parameters is derived in linear model. The superiorities of the BLMRE over ordinary least square estimator (LSE) is studied in terms of the mean square error matrix (MSEM) criterion and Pitman closeness (PC) criterion.     

固定效应部分线性变系数面板模型的快速有效估计

本文将最小二乘支持向量机(LSSVM) 和二次推断函数法(QIF) 相结合，为个体内具有相关结构的固定效应部分线性变系数面板模型提供了一种新的快速估计方法；在一定的正则条件下，论证了参数估计量的渐近正态性和非参数估计量的收敛速度；采用Monte Carlo模拟考察了估计方法在有限样本下的表现并将估计技术应用于现实数据分析。该方法不仅保证了估计的有效性和统计推断力，而且程序运行速度得到较大幅度提升。     

Empirical Likelihood Confidence Region for the Parameter in a Partially Linear Errors-in-Variables Model

In this article, a partially linear errors-in-variables model is considered, and empirical log-likelihood ratio statistic for the unknown parameter in the model is suggested. It is proved that the proposed statistic is asymptotically standard chi-square distribution under some suitable conditions, and hence it can be used to construct the confidence region of the parameter. A simulation study indicates that, in terms of coverage probabilities and average lengths of the confidence intervals, the proposed method performs better than the least-squares method.