Empirical likelihood for linear regression models under imputation for missing responses

The authors study the empirical likelihood method for linear regression models. They show that when missing responses are imputed using least squares predictors, the empirical log‐likelihood ratio is asymptotically a weighted sum of chi‐square variables with unknown weights. They obtain an adjusted empirical log‐likelihood ratio which is asymptotically standard chi‐square and hence can be used to construct confidence regions. They also obtain a bootstrap empirical log‐likelihood ratio and use its distribution to approximate that of the empirical log‐likelihood ratio. A simulation study indicates that the proposed methods are comparable in terms of coverage probabilities and average lengths of confidence intervals, and perform better than a normal approximation based method.     

Empirical likelihood for generalized linear models with missing responses

The paper uses the empirical likelihood method to study the construction of confidence intervals and regions for regression coefficients and response mean in generalized linear models with missing response. By using the inverse selection probability weighted imputation technique, the proposed empirical likelihood ratios are asymptotically chi-squared. Our approach is to directly calibrate the empirical likelihood ratio, which is called as a bias-correction method. Also, a class of estimators for the parameters of interest is constructed, and the asymptotic distributions of the proposed estimators are obtained. A simulation study indicates that the proposed methods are comparable in terms of coverage probabilities and average lengths/areas of confidence intervals/regions. An example of a real data set is used for illustrating our methods.     

Empirical likelihood inferences for varying coefficient partially nonlinear models

In this article, empirical likelihood inferences for the varying coefficient partially nonlinear models are investigated. An empirical log-likelihood ratio function for the unknown parameter vector in the nonlinear function part and a residual-adjusted empirical log-likelihood ratio function for the nonparametric component are proposed. The corresponding Wilks phenomena are proved and the confidence regions for parametric component and nonparametric component are constructed. Simulation studies indicate that, in terms of coverage probabilities and average areas of the confidence regions, the empirical likelihood method performs better than the normal approximation-based method. Furthermore, a real data set application is also provided to illustrate the proposed empirical likelihood estimation technique.     

Empirical Likelihood Inference for the Cox Model with Time-dependent Coefficients via Local Partial Likelihood

Abstract.  The Cox model with time-dependent coefficients has been studied by a number of authors recently. In this paper, we develop empirical likelihood (EL) pointwise confidence regions for the time-dependent regression coefficients via local partial likelihood smoothing. The EL simultaneous confidence bands for a linear combination of the coefficients are also derived based on the strong approximation methods. The EL ratio is formulated through the local partial log-likelihood for the regression coefficient functions. Our numerical studies indicate that the EL pointwise/simultaneous confidence regions/bands have satisfactory finite sample performances. Compared with the confidence regions derived directly based on the asymptotic normal distribution of the local constant estimator, the EL confidence regions are overall tighter and can better capture the curvature of the underlying regression coefficient functions. Two data sets, the gastric cancer data and the Mayo Clinic primary biliary cirrhosis data, are analysed using the proposed method.     

Generalized empirical likelihood inference in partial linear regression model for longitudinal data

In this paper, empirical likelihood inference for longitudinal data within the framework of partial linear regression models are investigated. The proposed procedures take into consideration the correlation within groups without involving direct estimation of nuisance parameters in the correlation matrix. The empirical likelihood method is used to estimate the regression coefficients and the baseline function, and to construct confidence intervals. A nonparametric version of Wilk's theorem for the limiting distribution of the empirical likelihood ratio is derived. Compared with methods based on normal approximations, the empirical likelihood does not require consistent estimators for the asymptotic variance and bias. The finite sample behaviour of the proposed method is evaluated with simulation and illustrated with an AIDS clinical trial data set.     

Empirical Likelihood Inferences for Semiparametric Varying-Coefficient Partially Linear Models with Longitudinal Data

In this article, empirical likelihood inferences for semiparametric varying-coefficient partially linear models with longitudinal data are investigated. We propose a groupwise empirical likelihood procedure to handle the inter-series dependence of the longitudinal data. By using residual-adjustment, an empirical likelihood ratio function for the nonparametric component is constructed, and a nonparametric version Wilks' phenomenons is proved. Compared with methods based on normal approximations, the empirical likelihood does not require consistent estimators for the asymptotic variance and bias. A simulation study is undertaken to assess the finite sample performance of the proposed confidence regions.     

Weighted composite quantile regression method via empirical likelihood for non linear models

In this paper, we investigate empirical likelihood (EL) inferences via weighted composite quantile regression for non linear models. Under regularity conditions, we establish that the proposed empirical log-likelihood ratio is asymptotically chi-squared, and then the confidence intervals for the regression coefficients are constructed. The proposed method avoids estimating the unknown error density function involved in the asymptotic covariance matrix of the estimators. Simulations suggest that the proposed EL procedure is more efficient and robust, and a real data analysis is used to illustrate the performance.     

Penalized empirical likelihood inference for sparse additive hazards regression with a diverging number of covariates

High-dimensional sparse modeling with censored survival data is of great practical importance, as exemplified by applications in high-throughput genomic data analysis. In this paper, we propose a class of regularization methods, integrating both the penalized empirical likelihood and pseudoscore approaches, for variable selection and estimation in sparse and high-dimensional additive hazards regression models. When the number of covariates grows with the sample size, we establish asymptotic properties of the resulting estimator and the oracle property of the proposed method. It is shown that the proposed estimator is more efficient than that obtained from the non-concave penalized likelihood approach in the literature. Based on a penalized empirical likelihood ratio statistic, we further develop a nonparametric likelihood approach for testing the linear hypothesis of regression coefficients and constructing confidence regions consequently. Simulation studies are carried out to evaluate the performance of the proposed methodology and also two real data sets are analyzed.     

Empirical Likelihood for Linear Models Under Linear Process Errors

In this article, we study the construction of confidence intervals for regression parameters in a linear model under linear process errors by using the blockwise technique. It is shown that the blockwise empirical likelihood (EL) ratio statistic is asymptotically χ²-type distributed. The result is used to obtain EL based confidence regions for regression parameters. The finite-sample performance of the method is evaluated through a simulation study.     

Two-sample high-dimensional empirical likelihood

In this paper, we apply empirical likelihood for two-sample problems with growing high dimensionality. Our results are demonstrated for constructing confidence regions for the difference of the means of two p-dimensional samples and the difference in value between coefficients of two p-dimensional sample linear model. We show that empirical likelihood based estimator has the efficient property. That is, as p → ∞ for high-dimensional data, the limit distribution of the EL ratio statistic for the difference of the means of two samples and the difference in value between coefficients of two-sample linear model is asymptotic normal distribution. Furthermore, empirical likelihood (EL) gives efficient estimator for regression coefficients in linear models, and can be as efficient as a parametric approach. The performance of the proposed method is illustrated via numerical simulations.     

Empirical likelihood for parameters in an additive partially linear errors-in-variables model with longitudinal data

Empirical likelihood inferences for the parameter component in an additive partially linear errors-in-variables model with longitudinal data are investigated in this article. A corrected-attenuation block empirical likelihood procedure is used to estimate the regression coefficients, a corrected-attenuation block empirical log-likelihood ratio statistic is suggested and its asymptotic distribution is obtained. Compared with the method based on normal approximations, our proposed method does not require any consistent estimator for the asymptotic variance and bias. Simulation studies indicate that our proposed method performs better than the method based on normal approximations in terms of relatively higher coverage probabilities and smaller confidence regions. Furthermore, an example of an air pollution and health data set is used to illustrate the performance of the proposed method.     

Empirical likelihood inference in mixture of semiparametric varying-coefficient models for longitudinal data with non-ignorable dropout

In this paper, empirical likelihood inference in mixture of semiparametric varying-coefficient models for longitudinal data with non-ignorable dropout is investigated. We estimate the non-parametric function based on the estimating equations and the local linear profile-kernel method. An empirical log-likelihood ratio statistic for parametric components is proposed to construct confidence regions and is shown to be an asymptotically chi-squared distribution. The non-parametric version of Wilk's theorem is also derived. A simulation study is undertaken to illustrate the finite sample performance of the proposed method.     

Empirical likelihood for a partially linear model with covariate data missing at random

In this paper, we consider the partial linear model with the covariables missing at random. Empirical likelihood ratios for the regression coefficients and the baseline function are investigated, the empirical log-likelihood ratios are proven to be asymptotically chi-squared and the corresponding confidence regions for the parameters of interest are then constructed. The finite sample behavior of the proposed method is evaluated with simulation and illustrated with an AIDS clinical trial dataset.     

Empirical Likelihood Based Synthetic Data Method for Censored Regression Analysis

In this article, we propose a new empirical likelihood method for linear regression analysis with a right censored response variable. The method is based on the synthetic data approach for censored linear regression analysis. A log-empirical likelihood ratio test statistic for the entire regression coefficients vector is developed and we show that it converges to a standard chi-squared distribution. The proposed method can also be used to make inferences about linear combinations of the regression coefficients. Moreover, the proposed empirical likelihood ratio provides a way to combine different normal equations derived from various synthetic response variables. Maximizing this empirical likelihood ratio yields a maximum empirical likelihood estimator which is asymptotically equivalent to the solution of the estimating equation that are optimal linear combination of the original normal equations. It improves the estimation efficiency. The method is illustrated by some Monte Carlo simulation studies as well as a real example.     

Empirical Likelihood Local Polynomial Regression Analysis of Clustered Data

Abstract. In this article, a naive empirical likelihood ratio is constructed for a non‐parametric regression model with clustered data, by combining the empirical likelihood method and local polynomial fitting. The maximum empirical likelihood estimates for the regression functions and their derivatives are obtained. The asymptotic distributions for the proposed ratio and estimators are established. A bias‐corrected empirical likelihood approach to inference for the parameters of interest is developed, and the residual‐adjusted empirical log‐likelihood ratio is shown to be asymptotically chi‐squared. These results can be used to construct a class of approximate pointwise confidence intervals and simultaneous bands for the regression functions and their derivatives. Owing to our bias correction for the empirical likelihood ratio, the accuracy of the obtained confidence region is not only improved, but also a data‐driven algorithm can be used for selecting an optimal bandwidth to estimate the regression functions and their derivatives. A simulation study is conducted to compare the empirical likelihood method with the normal approximation‐based method in terms of coverage accuracies and average widths of the confidence intervals/bands. An application of this method is illustrated using a real data set.     

Two Adjusted Empirical-Likelihood-Based Methods in Generalized Varying-Coefficient Partially Linear Model

An empirical likelihood method was proposed by Owen and has been extended to many semiparametric and nonparametric models with a continuous response variable. However, there has been less attention focused on the generalized regression model. This article systematically studies two adjusted empirical-likelihood-based methods in the generalized varying-coefficient partially linear models. Based on the popular profile likelihood estimation procedure, the new adjusted empirical likelihood technology for the parameter is established and the resulting statistics are shown to be asymptotically standard chi-square distributed. Further, the adjusted empirical-likelihood-based confidence regions are established, and an efficient adjusted profile empirical-likelihood-based confidence intervals/regions for any components of the parameter, which are of primary interest, is also constructed. Their asymptotic properties are also derived. Some numerical studies are carried out to illustrate the performance of the proposed inference procedures.     

Empirical Likelihood Based Rank Regression Inference

Rank regression procedures have been proposed and studied for numerous research applications that do not satisfy the underlying assumptions of the more common linear regression models. This article develops confidence regions for the slope parameter of rank regression using an empirical likelihood (EL) ratio method. It has the advantage of not requiring variance estimation which is required for the normal approximation method. The EL method is also range respecting and results in asymmetric confidence intervals. Simulation studies are used to compare and evaluate normal approximation versus EL inference methods for various conditions such as different sample size or error distribution. The simulation study demonstrates our proposed EL method almost outperforms the traditional method in terms of coverage probability, lower-tail side error, and upper-tail side error. An application of stability analysis also shows the EL method results in shorter confidence intervals for real life data.     

Bias-corrected estimations in varying-coefficient partially nonlinear models with measurement error in the nonparametric part

In this paper, we consider the statistical inference for the varying-coefficient partially nonlinear model with additive measurement errors in the nonparametric part. The local bias-corrected profile nonlinear least-squares estimation procedure for parameter in nonlinear function and nonparametric function is proposed. Then, the asymptotic normality properties of the resulting estimators are established. With the empirical likelihood method, a local bias-corrected empirical log-likelihood ratio statistic for the unknown parameter, and a corrected and residual adjusted empirical log-likelihood ratio for the nonparametric component are constructed. It is shown that the resulting statistics are asymptotically chi-square distribution under some suitable conditions. Some simulations are conducted to evaluate the performance of the proposed methods. The results indicate that the empirical likelihood method is superior to the profile nonlinear least-squares method in terms of the confidence regions of parameter and point-wise confidence intervals of nonparametric function.     

Empirical likelihood for density-weighted average derivatives

In this paper, we investigate empirical likelihood (EL) inference for density-weighted average derivatives in nonparametric multiple regression models. A simply adjusted empirical log-likelihood ratio for the vector of density-weighted average derivatives is defined and its limiting distribution is shown to be a standard Chi-square distribution. To increase the accuracy and coverage probability of confidence regions, an EL inference procedure for the rescaled parameter vector is proposed by using a linear instrumental variables regression. The new method shares the same properties of the regular EL method with i.i.d. samples. For example, estimation of limiting variances and covariances is not needed. A Monte Carlo simulation study is presented to compare the new method with the normal approximation method and an existing EL method.     

Empirical likelihood for high-dimensional partially linear model with martingale difference errors

In this paper, we focus on the empirical likelihood (EL) inference for high-dimensional partially linear model with martingale difference errors. An empirical log-likelihood ratio statistic of unknown parameter is constructed and is shown to have asymptotically normality distribution under some suitable conditions. This result is different from those derived before. Furthermore, an empirical log-likelihood ratio for a linear combination of unknown parameter is also proposed and its asymptotic distribution is chi-squared. Based on these results, the confidence regions both for unknown parameter and a linear combination of parameter can be obtained. A simulation study is carried out to show that our proposed approach performs better than normal approximation-based method.