Empirical Likelihood Inference for Generalized Partially Linear Models with Longitudinal Data

In this article, we propose a generalized empirical likelihood inference for the parametric component in semiparametric generalized partially linear models with longitudinal data. Based on the extended score vector, a generalized empirical likelihood ratios function is defined, which integrates the within-cluster?correlation meanwhile avoids direct estimating the nuisance parameters in the correlation matrix. We show that the proposed statistics are asymptotically?Chi-squared under some suitable conditions, and hence it can be used to construct the confidence region of parameters. In addition, the maximum empirical likelihood estimates of parameters and the corresponding asymptotic normality are obtained. Simulation studies demonstrate the performance of the proposed method.

Variable Selection of Partially Linear Single-index Models

In this article, we study the variable selection of partially linear single-index model(PLSIM). Based on the minimized average variance estimation, the variable selection of PLSIM is done by minimizing average variance with adaptive l1 penalty. Implementation algorithm is given. Under some regular conditions, we demonstrate the oracle properties of aLASSO procedure for PLSIM. Simulations are used to investigate the effectiveness of the proposed method for variable selection of PLSIM.

Sieve MLE for Generalized Partial Linear Models with Type Ⅱ Interval-censored Data

This article concerded with a semiparametric generalized partial linear model （GPLM） with the type Ⅱ censored data. A sieve maximum likelihood estimator （MLE） is proposed to estimate the parameter component, allowing exploration of the nonlinear relationship between a certain covariate and the response function. Asymptotic properties of the proposed sieve MLEs are discussed. Under some mild conditions, the estimators are shown to be strongly consistent. Moreover, the estimators of the unknown parameters are asymptotically normal and efficient, and the estimator of the nonparametric function has an optimal convergence rate.

ASYMPTOTIC PROPERTIES OF ESTIMATORS IN PARTIALLY LINEAR SINGLE-INDEX MODEL FOR LONGITUDINAL DATA

In this article, a partially linear single-index model /or longitudinal data is investigated. The generalized penalized spline least squares estimates of the unknown parameters are suggested. All parameters can be estimated simultaneously by the proposed method while the feature of longitudinal data is considered. The existence, strong consistency and asymptotic normality of the estimators are proved under suitable conditions. A simulation study is conducted to investigate the finite sample performance of the proposed method. Our approach can also be used to study the pure single-index model for longitudinal data.

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.

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.

Variable Selection for Generalized Linear Model with Highly Correlated Covariates

The penalized variable selection methods are often used to select the relevant covariates and estimate the unknown regression coefficients simultaneously,but these existing methods may fail to be consistent for the setting with highly correlated covariates.In this paper,the semi-standard partial covariance(SPAC)method with Lasso penalty is proposed to study the generalized linear model with highly correlated covariates,and the consistencies of the estimation and variable selection are shown in high-dimensional settings under some regularity conditions.Some simulation studies and an analysis of colon tumor dataset are carried out to show that the proposed method performs better in addressing highly correlated problem than the traditional penalized variable selection methods.

ROBUST ESTIMATION IN PARTIAL LINEAR MIXED MODEL FOR LONGITUDINAL DATA

In this article, robust generalized estimating equation for the analysis of partial linear mixed model for longitudinal data is used. The authors approximate the nonparametric function by a regression spline. Under some regular conditions, the asymptotic properties of the estimators are obtained. To avoid the computation of high-dimensional integral, a robust Monte Carlo Newton-Raphson algorithm is used. Some simulations are carried out to study the performance of the proposed robust estimators. In addition, the authors also study the robustness and the efficiency of the proposed estimators by simulation. Finally, two real longitudinal data sets are analyzed.

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.

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.

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.

Empirical likelihood confidence regions of the parameters in a partially linear single-index model

In this paper, a partially linear single-index model is investigated, and three empirical log-likelihood ratio statistics for the unknown parameters in the model are suggested. It is proved that the proposed statistics are asymptotically standard chi-square under some suitable conditions, and hence can be used to construct the confidence regions of the parameters. Our methods can also deal with the confidence region construction for the index in the pure single-index model. A simulation study indicates that, in terms of coverage probabilities and average areas of the confidence regions, the proposed methods perform better than the least-squares method.

Statistical inference on parametric part for partially linear single-index model

Statistical inference on parametric part for the partially linear single-index model (PLSIM) is considered in this paper. A profile least-squares technique for estimating the parametric part is proposed and the asymptotic normality of the profile least-squares estimator is given. Based on the estimator, a generalized likelihood ratio (GLR) test is proposed to test whether parameters on linear part for the model is under a contain linear restricted condition. Under the null model, the proposed GLR statistic follows asymptotically the χ2-distribution with the scale constant and degree of freedom independent of the nuisance parameters, known as Wilks phenomenon. Both simulated and real data examples are used to illustrate our proposed methods.

Tests for nonparametric parts on partially linear single index models

Tests for nonparametric parts on partially linear single index models are considered in this paper. Based on the estimates obtained by the local linear method, the generalized likelihood ratio tests for the models are established. Under the null hypotheses the normalized tests follow asymptotically the χ2-distribution with the scale constants and the degrees of freedom being independent of the nuisance parameters, which is called the Wilks phenomenon. A simulated example is used to evaluate the performances of the testing procedures empirically.

Delete-group Jackknife Estimate in Partially Linear Regression Models with Heteroscedasticity

Consider a partially linear regression model with an unknown vector parameter , an unknown function g(·), and unknown heteroscedastic error variances. Chen, You[23] proposed a semiparametric generalized least squares estimator (SGLSE) for , which takes the heteroscedasticity into account to increase efficiency. For inference based on this SGLSE, it is necessary to construct a consistent estimator for its asymptotic covariance matrix. However, when there exists within-group correlation, the traditional delta method and the delete-1 jackknife estimation fail to offer such a consistent estimator. In this paper, by deleting grouped partial residuals a delete-group jackknife method is examined. It is shown that the delete-group jackknife method indeed can provide a consistent estimator for the asymptotic covariance matrix in the presence of within-group correlations. This result is an extension of that in [21].

Testing Serial Correlation in Semiparametric Varying-Coefficient Partially Linear EV Models

This paper studies estimation and serial correlation test of a semiparametric varying-coefficient partially linear EV model of the form Y = X^Tβ ＋Z^Tα（T） ＋ε,ξ = X ＋ η with the identifying condition E[（ε,η^T）^T] =0, Cov[（ε,η^T）^T] = σ^2Ip＋1. The estimators of interested regression parameters /3 , and the model error variance σ2, as well as the nonparametric components α（T）, are constructed. Under some regular conditions, we show that the estimators of the unknown vector β and the unknown parameter σ2 are strongly consistent and asymptotically normal and that the estimator of α（T） achieves the optimal strong convergence rate of the usual nonparametric regression. Based on these estimators and asymptotic properties, we propose the VN,p test statistic and empirical log-likelihood ratio statistic for testing serial correlation in the model. The proposed statistics are shown to have asymptotic normal or chi-square distributions under the null hypothesis of no serial correlation. Some simulation studies are conducted to illustrate the finite sample performance of the proposed tests.

Partially Linear Single-Index Model in the Presence of Measurement Error

The partially linear single-index model(PLSIM) is a flexible and powerful model for analyzing the relationship between the response and the multivariate covariates. This paper considers the PLSIM with measurement error possibly in all the variables. The authors propose a new efficient estimation procedure based on the local linear smoothing and the simulation-extrapolation method,and further establish the asymptotic normality of the proposed estimators for both the index parameter and nonparametric link function. The authors also carry out extensive Monte Carlo simulation studies to evaluate the finite sample performance of the new method, and apply it to analyze the osteoporosis prevention data.

ZERO FINITE-ORDER SERIAL CORRELATION TEST IN A PARTIALLY LINEAR SINGLE-INDEX MODEL

The purpose of this paper is to test the underlying serial correlation in a partially linear single-index model. Under mild conditions, the proposed test statistics are shown to have standard chi- squared distribution asymptotically when there is no serial correlation in the error terms. To illustrate their finite sample properties, simulation experiments, as well as a real data example, are also provided. It is revealed that the finite sample performances of the proposed test statistics are satisfactory in terms of both estimated sizes and powers.

Inference on Varying-Coefficient Partially Linear Regression Model

The varying-coefficient partially linear regression model is proposed by combining nonparametric and varying-coefficient regression procedures. Wong, et al. （2008） proposed the model and gave its estimation by the local linear method. In this paper its inference is addressed. Based on these estimates, the generalized like- lihood ratio test is established. Under the null hypotheses the normalized test statistic follows a x2-distribution asymptotically, with the scale constant and the degrees of freedom being independent of the nuisance param- eters. This is the Wilks phenomenon. Furthermore its asymptotic power is also derived, which achieves the optimal rate of convergence for nonparametric hypotheses testing. A simulation and a real example are used to evaluate the performances of the testing procedures empirically.

基于GPLM的40～65岁绝经后骨质疏松症风险判别模型分析

目的建立基于广义偏线性模型(generalized partial linear model,GPLM)的,包括危险因素和中医证候要素内容的绝经后骨质疏松症(postmenopausal osteoporosis,PMOP)风险判别模型。方法在获取1740例社区PMOP高危人群危险因素及证候问卷调查数据基础上,筛选出与PMOP发病相关的重要危险因素和中医症状为协变量,以骨密度定性诊断为结局变量,建立基于GPLM的PMOP判别模型。结果 GPLM模型线性部分参数估计提示:是否绝经、体重指数、下肢抽筋、下肢骨痛、绝经年限(线性效应)具有统计意义(P<0.05);模型非线性部分参数估计提示:绝经年限(非线性效应)具有统计意义(P<0.05)。与logistic回归模型相比,拟合GPLM模型时加入了"绝经年限"的非线性效应,其AUC值为0.7971,具有统计学意义(χ2=21.9162,P<0.001)。结论绝经年限与PMOP发病之间存在非线性效应。将西医危险因素和中医症状相结合,建立基于GPLM的PMOP判别模型,反映病证结合特点,与logistic回归模型相比,具有更好的判别准确性。