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.     

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.     

A Comparative Study of Semiparametric Estimation in Partially Linear Single-index Models

We consider a semiparametric method based on partial splines for estimating the unknown function and partially linear regression parameters in partially linear single-index models. Three methods—project pursuit regression (PPR), average derivative estimation (ADE), and a boosting method—are considered for estimating the single-index parameters. Simulations revealed that PPR with partial splines was superior in estimating single-index parameters, while the boosting method with partial splines performed no better than PPR and ADE. All three methods performed similarly in estimating the partially linear regression parameters. The relative performances of the methods are also illustrated using a real-world data example.     

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。     

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.     

Efficient Estimation in Heteroscedastic Partially Linear Varying Coefficient Models

This article considers statistical inference for the heteroscedastic partially linear varying coefficient models. We construct an efficient estimator for the parametric component by applying the weighted profile least-squares approach, and show that it is semiparametrically efficient in the sense that the inverse of the asymptotic variance of the estimator reaches the semiparametric efficiency bound. Simulation studies are conducted to illustrate the performance of the proposed method.     

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.     

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.     

Comparison of Nonparametric Components in Two Partially Linear Models

In this article, we are concerned with whether the nonparametric functions are parallel from two partial linear models, and propose a test statistic to check the difference of the two functions. The unknown constant α is estimated by using moment method under null models. Nonparametric functions under both null and full models are estimated by using local linear method. The asymptotic properties of parametric and nonparametric components are derived. The test statistic under the null hypothesis is calculated and shown to be asymptotically normal.     

Semiparametric Estimation of Stochastic Production Frontier Models

This article extends the linear stochastic frontier model proposed by Aigner, Lovell, and Schmidt to a semiparametric frontier model in which the functional form of the production frontier is unspecified and the distributions of the composite error terms are of known form. Pseudolikelihood estimators of the parameters characterizing the two error terms of the model are constructed based on kernel estimation of the conditional mean function. The Monte Carlo results show that the proposed estimators perform well in finite samples. An empirical application is presented. Extensions to a partially linear frontier function and to more flexible one-sided error distributions than the half-normal are discussed     

Estimation for Partially Linear Single-index Instrumental Variables Models

In this article, we generalize the partially linear single-index models to the scenario with some endogenous covariates variables. It is well known that the estimators based on the existing methods are often inconsistent because of the endogeneity of covariates. To deal with the endogenous variables, we introduce some auxiliary instrumental variables. A three-stage estimation procedure is proposed for partially linear single-index instrumental variables models. The first stage is to obtain a linear projection of endogenous variables on a set of instrumental variables, the second stage is to estimate the link function by using local linear smoother for given constant parameters, and the last stage is to obtain the estimators of constant parameters based on the estimating equation. Asymptotic normality is established for the proposed estimators. Some simulation studies are undertaken to assess the finite sample performance of the proposed estimation procedure.     

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.     

Empirical Likelihood for Semiparametric Varying-Coefficient Heteroscedastic Partially Linear Errors-in-Variables Models

The purpose of this article is to use the empirical likelihood method to study construction of the confidence region for the parameter of interest in semiparametric varying-coefficient heteroscedastic partially linear errors-in-variables models. When the variance functions of the errors are known or unknown, we propose the empirical log-likelihood ratio statistics for the parameter of interest. For each case, a nonparametric version of Wilks’ theorem is derived. The results are then used to construct confidence regions of the parameter. A simulation study is carried out to assess the performance of the empirical likelihood method.     

A Semiparametric Estimation Approach for Linear Mixed Models

Maximum likelihood approach is the most frequently employed approach for the inference of linear mixed models. However, it relies on the normal distributional assumption of the random effects and the within-subject errors, and it is lack of robustness against outliers. This article proposes a semiparametric estimation approach for linear mixed models. This approach is based on the first two marginal moments of the response variable, and does not require any parametric distributional assumptions of random effects or error terms. The consistency and asymptotically normality of the estimator are derived under fairly general conditions. In addition, we show that the proposed estimator has a bounded influence function and a redescending property so it is robust to outliers. The methodology is illustrated through an application to the famed Framingham cholesterol data. The finite sample behavior and the robustness properties of the proposed estimator are evaluated through extensive simulation studies.     

Robust Tests in Semiparametric Partly Linear Models

Abstract.  This paper focuses on the problem of testing the null hypothesis that the regression parameter equals a fixed value under a semiparametric partly linear regression model by using a three-step robust estimate for the regression parameter and the regression function. Two families of tests statistics are considered and their asymptotic distributions are studied under the null hypothesis and under contiguous alternatives. A Monte Carlo study is performed to compare the finite sample behaviour of the proposed tests with the classical one.     

Testing Serial Correlation in Partially Linear Single-Index Errors-in-Variables Models

In this article, we propose two test statistics for testing the underlying serial correlation in a partially linear single-index model Y = η(Z ^τα) + X ^τβ + ? when X is measured with additive error. The proposed test statistics are shown to have asymptotic normal or chi-squared distributions under the null hypothesis of no serial correlation. Monte Carlo experiments are also conducted to illustrate the finite sample performance of the proposed test statistics. The simulation results confirm that these statistics perform satisfactorily in both estimated sizes and powers.     

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.     

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.     

Testing Serial Correlation in Semiparametric Varying-Coefficient Partially Linear Models

We propose two test statistics for testing serial correlation in semiparametric varying-coefficient partially linear models. The proposed test statistics are not only for testing zero first-order serial correlation, but also for testing higher-order serial correlations. Under the null hypothesis of no serial correlation, the test statistics are shown to have asymptotic normal or chi-square distributions. By using R, some Monte Carlo experiments are conducted to examine the finite sample performances of the proposed tests. Simulation results show that the estimated size and power of the proposed tests behave well.     

部分线性模型非参数部分的多项式类关系的检验

对于部分线性模型中非参数部分是否为多项式函数的检验问题,应该先确定其是否为多项式函数类。通过对部分线性模型的拟合残差进行再光滑,基于其变化的趋势性构造统计量以检验其是否为多项式函数类,给出了计算检验P-值的精确算法和三阶矩χ2逼近方法,模拟例子与实际例子充分显示了本方法的有效性。