Empirical Likelihood for the Additive Hazards Model with Current Status Data

We investigate empirical likelihood for the additive hazards model with current status data. An empirical log-likelihood ratio for a vector or subvector of regression parameters is defined and its limiting distribution is shown to be a standard chi-squared distribution. The proposed inference procedure enables us to make empirical likelihood-based inference for the regression parameters. Finite sample performance of the proposed method is assessed in simulation studies to compare with that of a normal approximation method, it shows that the empirical likelihood method provides more accurate inference than the normal approximation method. A real data example is used for illustration.     

The Empirical Likelihood Inference of a Regression Parameter in Censored Partial Linear Models Based on a Piecewise Polynomial

This article aims at making an empirical likelihood inference of regression parameter in partial linear model when the response variable is right censored randomly. The present studies are mainly designed to use empirical likelihood (EL) method based on synthetic dependent data, and the result cannot be applied directly due to the unknown weights in it. In this paper, we introduce a censored empirical log-likelihood ratio and demonstrate that its limiting distribution is a standard chi-square distribution. The estimating procedure of β is developed based on piecewise polynomial method. As a result, the p-value of test and the confidence interval can be obtained without estimating other quantities. Some simulation studies are conducted to highlight the performance of the proposed EL method, and the results show a good performance. Finally, we apply our method into the real example of multiple myeloma data and show the proof of theorem.     

Empirical Likelihood Inference of the Partial Linear Isotonic Errors-in-variables Regression Models with Missing Data

This article is concerned with statistical inference of the partial linear isotonic regression model missing response and measurement errors in covariates. We proposed an empirical likelihood ratio test statistics and show that it has a limiting weighted chi-square distribution. An adjusted empirical likelihood ratio statistic, which is shown to have a limiting standard central chi-square distribution, is then proposed further. A maximum empirical likelihood estimator is also developed. A simulation study is conducted to examine the finite-sample property of proposed procedure.     

Inference for the Mean Residual Life Function via Empirical Likelihood

In addition to the distribution function, the mean residual life (MRL) function is the other important function which can be used to characterize a lifetime in survival analysis and reliability. For inference on the MRL function, some procedures have been proposed in the literature. However, the coverage accuracy of such procedures may be low when the sample size is small. In this article, an empirical likelihood (EL) inference procedure of MRL function is proposed and the limiting distribution of the EL ratio for MRL function is derived. Based on the result, we obtain confidence interval/band for the MRL function. The proposed method is compared with the normal approximation based method through simulation study in terms of coverage probability.     

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.     

Semiparametric estimation for weighted average derivatives with responses missing at random

When responses are missing at random, we propose a semiparametric direct estimator for the missing probability and density-weighted average derivatives of a general nonparametric multiple regression function. An estimator for the normalized version of the weighted average derivatives is constructed as well using instrumental variables regression. The proposed estimators are computationally simple and asymptotically normal, and provide a solution to the problem of estimating index coefficients of single-index models with responses missing at random. The developed theory generalizes the method of the density-weighted average derivatives estimation of Powell et al. (1989) for the non-missing data case. Monte Carlo simulation studies are conducted to study the performance of the methods.     

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.     

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.     

Weighted empirical likelihood for generalized linear models with longitudinal data

In this paper, we introduce the empirical likelihood (EL) method to longitudinal studies. By considering the dependence within subjects in the auxiliary random vectors, we propose a new weighted empirical likelihood (WEL) inference for generalized linear models with longitudinal data. We show that the weighted empirical likelihood ratio always follows an asymptotically standard chi-squared distribution no matter which working weight matrix that we have chosen, but a well chosen working weight matrix can improve the efficiency of statistical inference. Simulations are conducted to demonstrate the accuracy and efficiency of our proposed WEL method, and a real data set is used to illustrate 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.     

Generalized empirical likelihood inference in partially linear model for longitudinal data with missing response variables and error-prone covariates

In this article, we consider statistical inference for longitudinal partial linear models when the response variable is sometimes missing with missingness probability depending on the covariate that is measured with error. A generalized empirical likelihood (GEL) method is proposed by combining correction attenuation and quadratic inference functions. The method that takes into consideration the correlation within groups is used to estimate the regression coefficients. Furthermore, residual-adjusted empirical likelihood (EL) is employed for estimating the baseline function so that undersmoothing is avoided. 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. Compared with methods based on NAs, the GEL does not require consistent estimators for the asymptotic variance and bias. The numerical study is conducted to compare the performance of the EL and the normal approximation-based method, and a real example is analysed.     

Empirical Likelihood For Censored Partial Linear Model Based On Imputed Value

This article aims at proposing a new type of empirical likelihood testing procedure based on the Wilks theorem and imputed value in censored partial linear model. The present study is mainly designed to use empirical likelihood (EL) method based on synthetic dependent data, and the result can not be applied directly due to the weights in it. In this article, a censored empirical log-likelihood ratio is introduced to tackle this problem. Particularly, we demonstrate that its limiting distribution is a standard chi-squared distribution with freedom of one. This method is used to calculate the p-value and construct the confidence interval. Some simulation studies are conducted to highlight the performance of the proposed EL method, and the results show that it performs well. Finally, an illustration is given using the Stanford Heart Transplant data.     

Penalized empirical likelihood for quantile regression with missing covariates and auxiliary information

Based on the inverse probability weight method, we, in this article, construct the empirical likelihood (EL) and penalized empirical likelihood (PEL) ratios of the parameter in the linear quantile regression model when the covariates are missing at random, in the presence and absence of auxiliary information, respectively. It is proved that the EL ratio admits a limiting Chi-square distribution. At the same time, the asymptotic normality of the maximum EL and PEL estimators of the parameter is established. Also, the variable selection of the model in the presence and absence of auxiliary information, respectively, is discussed. Simulation study and a real data analysis are done to evaluate the performance of the proposed methods.     

Empirical Likelihood Method for General Additive-Multiplicative Hazard Models

An empirical likelihood-based inferential procedure is developed for a class of general additive-multiplicative hazard models. The proposed log-empirical likelihood ratio test statistic for the parameter vector is shown to have a chi-squared limiting distribution. The result can be used to make inference about the entire parameter vector as well as any linear combination of it. The asymptotic power of the proposed test statistic under contiguous alternatives is discussed. The method is illustrated by extensive simulation studies and a real example.     

Adjusted empirical likelihood inference for additive hazards regression

ABSTRACT

This article develops an adjusted empirical likelihood (EL) method for the additive hazards model. The adjusted EL ratio is shown to have a central chi-squared limiting distribution under the null hypothesis. We also evaluate its asymptotic distribution as a non central chi-squared distribution under the local alternatives of order n^{? 1/2}, deriving the expression for the asymptotic power function. Simulation studies and a real example are conducted to evaluate the finite sample performance of the proposed method. Compared with the normal approximation-based method, the proposed method tends to have more larger empirical power and smaller confidence regions with comparable coverage probabilities.     

Empirical likelihood-based inference in nonlinear regression models with missing responses at random

This paper investigates the estimations of regression parameters and response mean in nonlinear regression models in the presence of missing response variables that are missing with missingness probabilities depending on covariates. We propose four empirical likelihood (EL)-based estimators for the regression parameters and the response mean. The resulting estimators are shown to be consistent and asymptotically normal under some general assumptions. To construct the confidence regions for the regression parameters as well as the response mean, we develop four EL ratio statistics, which are proven to have the χ² distribution asymptotically. Simulation studies and an artificial data set are used to illustrate the proposed methodologies. Empirical results show that the EL method behaves better than the normal approximation method and that the coverage probabilities and average lengths depend on the selection probability function.     

Penalized Empirical Likelihood via Bridge Estimator in Cox's Proportional Hazard Model

We propose the penalized empirical likelihood method via bridge estimator in Cox's proportional hazard model for parameter estimation and variable selection. Under reasonable conditions, we show that penalized empirical likelihood in Cox's proportional hazard model has oracle property. A penalized empirical likelihood ratio for the vector of regression coefficients is defined and its limiting distribution is a chi-square distributions. The advantage of penalized empirical likelihood as a nonparametric likelihood approach is illustrated in testing hypothesis and constructing confidence sets. The method is illustrated by extensive simulation studies and a real example.     

Empirical phi-divergence test statistics for testing simple and composite null hypotheses

The main purpose of this paper is to introduce first a new family of empirical test statistics for testing a simple null hypothesis when the vector of parameters of interest is defined through a specific set of unbiased estimating functions. This family of test statistics is based on a distance between two probability vectors, with the first probability vector obtained by maximizing the empirical likelihood (EL) on the vector of parameters, and the second vector defined from the fixed vector of parameters under the simple null hypothesis. The distance considered for this purpose is the phi-divergence measure. The asymptotic distribution is then derived for this family of test statistics. The proposed methodology is illustrated through the well-known data of Newcomb's measurements on the passage time for light. A simulation study is carried out to compare its performance with that of the EL ratio test when confidence intervals are constructed based on the respective statistics for small sample sizes. The results suggest that the ‘empirical modified likelihood ratio test statistic’ provides a competitive alternative to the EL ratio test statistic, and is also more robust than the EL ratio test statistic in the presence of contamination in the data. Finally, we propose empirical phi-divergence test statistics for testing a composite null hypothesis and present some asymptotic as well as simulation results for evaluating the performance of these test procedures.     

Empirical likelihood inference for the regression model of mean quality‐adjusted lifetime with censored data

The authors consider the empirical likelihood method for the regression model of mean quality‐adjusted lifetime with right censoring. They show that an empirical log‐likelihood ratio for the vector of the regression parameters is asymptotically a weighted sum of independent chi‐squared random variables. They adjust this empirical log‐likelihood ratio so that the limiting distribution is a standard chi‐square and construct corresponding confidence regions. Simulation studies lead them to conclude that empirical likelihood methods outperform the normal approximation methods in terms of coverage probability. They illustrate their methods with a data example from a breast cancer clinical trial study.     

Empirical Likelihood for Censored Linear Regression

In this paper we investigate the empirical likelihood method in a linear regression model when the observations are subject to random censoring. An empirical likelihood ratio for the slope parameter vector is defined and it is shown that its limiting distribution is a weighted sum of independent chi-square distributions. This reduces to the empirical likelihood to the linear regression model first studied by Owen (1991) if there is no censoring present. Some simulation studies are presented to compare the empirical likelihood method with the normal approximation based method proposed in Lai et al. (1995). It was found that the empirical likelihood method performs much better than the normal approximation method.