Some Asymptotic Properties for Multivariate Partially Linear Models

The paper considers a multivariate partially linear model under independent errors,and investigates the asymptotic bias and variance-covariance for parametric component βand nonparametric component F（·）by the GJS estimator and Kernel estimation.

Testing Linearity of Nonparametric Component in Partially Linear Model

In this paper, we propose the test statistic to check whether the nonparametric function in partially linear models is linear or not. We estimate the nonparametric function in alternative by using the local linear method, and then estimate the parameters by the two stage method. The test statistic under the null hypothesis is calculated, and it is shown to be asymptotically normal.

The Application and Property of Elastic Net Procedure for Partially Linear Models

Variable selection plays an important role in high-dimensional data analysis.But the high-dimensional data often induces the strongly correlated variables problem,which should be properly handled.In this paper,we propose Elastic Net procedure for partially linear models and prove the group effect of its estimate.A simulation study shows that the Elastic Net procedure deals with the strongly correlated variables problem better than the Lasso,ALasso and the Ridge do.Based on the real world data study,we can get that the Elastic Net procedure is particularly useful when the number of predictors pffis much bigger than the sample size n.

Testing Equality of Nonparametric Functions in Two Partially Linear Models

We propose the test statistic to check whether the nonpararnetric functions in two partially linear models are equality or not in this paper. We estimate the nonparametric function both in null hypothesis and the alternative by the local linear method, where we ignore the parametric components, and then estimate the parameters by the two stage method. The test statistic is derived, and it is shown to be asymptotically normal under the null hypothesis.

ESTIMATORS AND SOME BEHAVIORS FORA PARTIALLY LINEAR MODEL WITH CENSORED DATA

This paper considers the local linear regression estimators for partially linear model with censored data. Which have some nice large-sample behaviors and are easy to implement. By many simulation runs, the author also found that the estimators show remarkable in the small sample case yet.

STRONG CONVERGENCE RATES OF SEVERAL ESTIMATORS IN SEMIPARAMETRIC VARYING-COEFFICIENT PARTIALLY LINEAR MODELS

This article is concerned with the estimating problem of semiparametric varyingcoefficient partially linear regression models. By combining the local polynomial and least squares procedures Fan and Huang （2005） proposed a profile least squares estimator for the parametric component and established its asymptotic normality. We further show that the profile least squares estimator can achieve the law of iterated logarithm. Moreover, we study the estimators of the functions characterizing the non-linear part as well as the error variance. The strong convergence rate and the law of iterated logarithm are derived for them, respectively.

Testing Linearity in Functional Partially Linear Models

For the functional partially linear models including flexible nonparametric part and functional linear part,the estimators of the nonlinear function and the slope function have been studied in existing literature.How to test the correlation between response and explanatory variables,however,still seems to be missing.Therefore,a test procedure for testing the linearity in the functional partially linear models will be proposed in this paper.A test statistic is constructed based on the existing estimators of the nonlinear and the slope functions.Further,we prove that the approximately asymptotic distribution of the proposed statistic is a chi-squared distribution under some regularity conditions.Finally,some simulation studies and a real data application are presented to demonstrate the performance of the proposed test statistic.

Empirical likelihood-based inference in a partially linear model for longitudinal data

A partially linear model with longitudinal data is considered, empirical likelihood to infer- ence for the regression coefficients and the baseline function is investigated, the empirical log-likelihood ratios is proven to be asymptotically chi-squared, and the corresponding confidence regions for the pa- rameters of interest are then constructed. Also by the empirical likelihood ratio functions, we can obtain the maximum empirical likelihood estimates of the regression coefficients and the baseline function, and prove the asymptotic normality. The numerical results are conducted to compare the performance of the empirical likelihood and the normal approximation-based method, and a real example is analysed.

Robust estimation for partially linear models with large-dimensional covariates

We are concerned with robust estimation procedures to estimate the parameters in partially linear models with large-dimensional covariates. To enhance the interpretability, we suggest implementing a noncon- cave regularization method in the robust estimation procedure to select important covariates from the linear component. We establish the consistency for both the linear and the nonlinear components when the covariate dimension diverges at the rate of o（√n）, where n is the sample size. We show that the robust estimate of linear component performs asymptotically as well as its oracle counterpart which assumes the baseline function and the unimportant covariates were known a priori. With a consistent estimator of the linear component, we estimate the nonparametric component by a robust local linear regression. It is proved that the robust estimate of nonlinear component performs asymptotically as well as if the linear component were known in advance. Comprehensive simulation studies are carried out and an application is presented to examine the finite-sample performance of the proposed procedures.

EMPIRICAL LIKELIHOOD FOR LONGITUDINAL PARTIALLY LINEAR MODEL WITH α-MIXING ERRORS

This paper considers large sample inference for the regression parameter in a partially linear regression model with longitudinal data and a-mixing errors. The authors introduce an estimated empirical likelihood for the regression parameter and show that its limiting distribution is a mixture of central chi-squared distributions. Also, the authors derive an adjusted empirical likelihood method which is shown to have a central chi-square limiting distribution. A simulation study is carried out to assess the performance of the empirical likelihood method.

Empirical Likelihood for Partially Linear Models Under Negatively Associated Errors

This paper proposes to use the blockwise empirical likelihood （EL） method to construct the confidence regions for the regression vector β in a partially linear model under negatively associated errors. It is shown that the blockwise EL ratio statistic for β is asymptotically χ^2 distributed. The result is used to obtain an EL-based confidence region for β. Results of a simulation study on the finite sample performance of the proposed confidence regions are reported.

Corrected-loss estimation for Error-in-Variable partially linear model

We consider an Error-in-Variable partially linear model where the covariates of linear part are measured with error which follows a normal distribution with a known covariance matrix. We propose a corrected-loss estimation of the covariate effect. The proposed estimator is asymptotically normal. Simulation studies are presented to show that the proposed method performs well with finite samples, and the proposed method is applied to a real data set.

Generalized F-Test for High Dimensional Regression Coefficients of Partially Linear Models

This paper proposes a test procedure for testing the regression coefficients in high dimensional partially linear models based on the F-statistic. In the partially linear model, the authors first estimate the unknown nonlinear component by some nonparametric methods and then generalize the F-statistic to test the regression coefficients under some regular conditions. During this procedure, the estimation of the nonlinear component brings much challenge to explore the properties of generalized F-test. The authors obtain some asymptotic properties of the generalized F-test in more general cases,including the asymptotic normality and the power of this test with p/n ∈(0, 1) without normality assumption. The asymptotic result is general and by adding some constraint conditions we can obtain the similar conclusions in high dimensional linear models. Through simulation studies, the authors demonstrate good finite-sample performance of the proposed test in comparison with the theoretical results. The practical utility of our method is illustrated by a real data example.

Empirical Likelihood Based Diagnostics for Heteroscedasticity in Semiparametric Varying-Coefficient Partially Linear Models with Missing Responses

This paper proposes an empirical likelihood based diagnostic technique for heteroscedasticity for semiparametric varying-coefficient partially linear models with missing responses. Firstly, the authors complement the missing response variables by regression method. Then, the empirical likelihood method is introduced to study the heteroscedasticity of the semiparametric varying-coefficient partially linear models with complete-case data. Finally, the authors obtain the finite sample property by numerical simulation.

Double Penalized Variable Selection Procedure for Partially Linear Models with Longitudinal Data

Based on the double penalized estimation method,a new variable selection procedure is proposed for partially linear models with longitudinal data.The proposed procedure can avoid the effects of the nonparametric estimator on the variable selection for the parameters components.Under some regularity conditions,the rate of convergence and asymptotic normality of the resulting estimators are established.In addition,to improve efficiency for regression coefficients,the estimation of the working covariance matrix is involved in the proposed iterative algorithm.Some simulation studies are carried out to demonstrate that the proposed method performs well.

Empirical Likelihood Test for Regression Coefficients in High Dimensional Partially Linear Models

This paper considers tests for regression coefficients in high dimensional partially linear Models.The authors first use the B-spline method to estimate the unknown smooth function so that it could be linearly expressed.Then,the authors propose an empirical likelihood method to test regression coefficients.The authors derive the asymptotic chi-squared distribution with two degrees of freedom of the proposed test statistics under the null hypothesis.In addition,the method is extended to test with nuisance parameters.Simulations show that the proposed method have a good performance in control of type-I error rate and power.The proposed method is also employed to analyze a data of Skin Cutaneous Melanoma(SKCM).

GENERALIZED p-VALUES FOR TESTING REGRESSION COEFFICIENTS IN PARTIALLY LINEAR MODELS

The one-sided and two-sided hypotheses about the parametric component in partially linear model are considered in this paper. Generalized p-values are proposed based on fiducial method for testing the two hypotheses at the presence of nonparametric nuisance parameter. Note that the nonparametric component can be approximated by a linear combination of some known functions, thus, the partially linear model can be approximated by a linear model. Thereby, generalized p-values for a linear model are studied first, and then the results are extended to the situation of partially linear model. Small sample frequency properties are analyzed theoretically. Meanwhile, simulations are conducted to assess the finite sample performance of the tests based on the proposed p-values.

Testing for Change Points in Partially Linear Models

In this paper we provide a method to test the existence of the change points in the nonparametric regression function of partially linear models with conditional heteroscedastic variance. We propose the test statistic and establish its asymptotic properties under some regular conditions. Some simulation studies are given to investigate the performance of the proposed method in finite samples. Finally, the proposed method is applied to a real data for illustration.

A Note on the Optimality of Generalized Cross-validation Bandwidth Selection in Partially Linear Models with Kernel Smoothing Estimator

The issue of selection of bandwidth in kernel smoothing method is considered within the context of partially linear models, hi this paper, we study the asymptotic behavior of the bandwidth choice based on generalized cross-validation （CCV） approach and prove that this bandwidth choice is asymptotically optimal. Numerical simulation are also conducted to investigate the empirical performance of generalized cross-valldation.

Estimation for Partially Linear Models with Missing Responses:the Fixed Design Case

Suppose that we have a partially linear model Yi = xiβ ＋ g（ti） ＋εi with independent zero mean errors εi, where （xi,ti, i = 1, ... ,n} are non-random and observed completely and （Yi, i = 1,...,n} are missing at random（MAR）. Two types of estimators of β and g（t） for fixed t are investigated： estimators based on semiparametric regression and inverse probability weighted imputations. Asymptotic normality of the estimators is established, which is used to construct normal approximation based confidence intervals on β and g（t）. Results are reported of a simulation study on the finite sample performance of the estimators and confidence intervals proposed in this paper.