Liugen Xue

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.

Efficient inference in a generalized partially linear model with random effect for longitudinal data

In this article, we study the statistical inference for the generalized partially linear model with random effect. We develop the traditional models that can model generalized longitudinal data and treat categorical data as continuous data by using some transformations. We propose a class of semiparametric estimators for the parametric and variance components. The proposed estimators are data adaptive, which does not require any assumption of working likelihood for the random component or model error. We prove that the resulting estimators for the parametric component are consistent and asymptotic normal, but also remain semiparametrically efficient. The asymptotic normality is established for the proposed estimator of variance component. Moreover, we also propose an estimator for the nonparametric component by using the local linear smoother and present their asymptotic normality. Finite sample performance of the proposed procedures is evaluated by Monte Carlo simulation studies. We further illustrate the proposed procedure by an application.

Generalized Empirical Likelihood Inference in Generalized Linear Models for Longitudinal Data

In this article, the generalized linear model for longitudinal data is studied. A generalized empirical likelihood method is proposed by combining generalized estimating equations and quadratic inference functions based on the working correlation matrix. It is proved that the proposed generalized empirical likelihood ratios are asymptotically chi-squared under some suitable conditions, and hence it can be used to construct the confidence regions of the parameters. In addition, the maximum empirical likelihood estimates of parameters are obtained, and their asymptotic normalities are proved. Some simulations are undertaken to compare the generalized empirical likelihood and normal approximation-based method in terms of coverage accuracies and average areas/lengths of confidence regions/intervals. An example of a real data is used for illustrating our methods.

Semiparametric estimation of the single-index varying-coefficient model

ABSTRACT In this paper, we consider the choice of pilot estimators for the single-index varying-coefficient model, which may result in radically different estimators, and develop the method for estimating the unknown parameter in this model. To estimate the unknown parameters efficiently, we use the outer product of gradient method to find the consistent initial estimators for interest parameters, and then adopt the refined estimation method to improve the efficiency, which is similar to the refined minimum average variance estimation method. An algorithm is proposed to estimate the model directly. Asymptotic properties for the proposed estimation procedure have been established. The bandwidth selection problem is also considered. Simulation studies are carried out to assess the finite sample performance of the proposed estimators, and efficiency comparisons between the estimation methods are made.

The Empirical Likelihood Goodness-of-Fit Test for a Regression Model with Randomly Censored Data

The regression model with randomly censored data has been intensively investigated. In this article, we consider a goodness-of-fit test for this model. Empirical likelihood (EL) tests are constructed. The asymptotic distributions of the test statistic under null hypothesis and the local alternative hypothesis are given. Simulations are carried out to illustrate the methodology.

Automatic Variable Selection for Partially Linear Functional Additive Model and Its Application to the Tecator Data Set

We introduce a new partially linear functional additive model, and we consider the problem of variable selection for this model. Based on the functional principal components method and the centered spline basis function approximation, a new variable selection procedure is proposed by using the smooth-threshold estimating equation (SEE). 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 SEE. The approach avoids the convex optimization problem, and it is flexible and easy to implement. We establish the asymptotic properties of the resulting estimators under some regularity conditions. We apply the proposed procedure to analyze a real data set: the Tecator data set.

Variable Selection for Varying-Coefficient Models with Missing Response at Random

ABSTRACT This paper focuses on the variable selections for a varying coefficient models with missing response at random. A procedure is presented by basis function approximations with smooth-threshold estimating equations. Furthermore, the proposed method selects significant variables and estimates coefficients simultaneously avoiding the problem of solving a convex optimization, which reduced the burden of computation. Compared to existing equation based approaches, our procedure is more efficient and quick. With proper choices the regularization parameter, the resulting estimates perform an oracle property. A cross-validation for tuning parameter selection is also proposed, a numerical study confirms the performance of the proposed method.

Variance estimation for sparse ultra-high dimensional varying coefficient models

ABSTRACT This paper considers the problem of variance estimation for sparse ultra-high dimensional varying coefficient models. We first use B-spline to approximate the coefficient functions, and discuss the asymptotic behavior of a naive two-stage estimator of error variance. We also reveal that this naive estimator may significantly underestimate the error variance due to the spurious correlations, which are even higher for nonparametric models than linear models. This prompts us to propose an accurate estimator of the error variance by effectively integrating the sure independence screening and the refitted cross-validation techniques. The consistency and the asymptotic normality of the resulting estimator are established under some regularity conditions. The simulation studies are carried out to assess the finite sample performance of the proposed methods.

Empirical likelihood inference for partial functional linear model with missing responses

ABSTRACT In this paper, we consider the empirical likelihood inferences of the partial functional linear model with missing responses. Two empirical log-likelihood ratios of the parameters of interest are constructed, and the corresponding maximum empirical likelihood estimators of parameters are derived. Under some regularity conditions, we show that the proposed two empirical log-likelihood ratios are asymptotic standard Chi-squared. Thus, the asymptotic results can be used to construct the confidence intervals/regions for the parameters of interest. We also establish the asymptotic distribution theory of corresponding maximum empirical likelihood estimators. A simulation study indicates that the proposed methods are comparable in terms of coverage probabilities and average lengths of confidence intervals. An example of real data is also used to illustrate our proposed methods.

CBPS-based estimation for linear models with responses missing at random

ABSTRACT In this article, based on the covariate balancing propensity score (CBPS), estimators for the regression coefficients and the population mean are obtained, when the responses of linear models are missing at random. It is proved that the proposed estimators are asymptotically normal. In simulation studies and real example, the proposed estimators show improved performance relative to usual augmented inverse probability weighted estimators.