Empirical Log-likelihood Ratio Statistic Research Articles

Robust estimation and bias-corrected empirical likelihood in generalized linear models with right censored data

In this paper, we study the robust estimation and empirical likelihood for the regression parameter in generalized linear models with right censored data. A robust estimating equation is proposed to estimate the regression parameter, and the resulting estimator has consistent and asymptotic normality. A bias-corrected empirical log-likelihood ratio statistic of the regression parameter is constructed, and it is shown that the statistic converges weakly to a standard χ 2 distribution. The result can be directly used to construct the confidence region of regression parameter. We use the bias correction method to directly calibrate the empirical log-likelihood ratio, which does not need to be multiplied by an adjustment factor. We also propose a method for selecting the tuning parameters in the loss function. Simulation studies show that the estimator of the regression parameter is robust and the bias-corrected empirical likelihood is better than the normal approximation method. An example of a real dataset from Alzheimer's disease studies shows that the proposed method can be applied in practical problems.

Empirical likelihood analysis for accelerated failure time model using length-biased data

Length-biased data arise when the probability of observing a subject in a sample is proportional to its corresponding value. The statistical tools that should be applied to analyse such sets of data differ from those used for random samples. Ignoring this issue in an investigation results in a biased inference. In this paper, we propose an empirical likelihood-based method to draw inference on covariate effects in an accelerated failure time model while the observations are subject to length-bias. The asymptotic distribution of the empirical log-likelihood ratio statistic is derived to be a weighted sum of independent chi-square distributions. Hence, we then extend the results by an exploration into the adjusted empirical likelihood. We derive an asymptotic standard chi-square distribution for the adjusted log-likelihood statistic. The limiting distributions are applied to obtain confidence regions for the regression parameters. A simulation study is carried out to evaluate and compare the performance of the proposed procedures with an existing method based on the normal approximation approach. Finally, the procedures are illustrated by modelling the regression parameter and estimating confidence intervals for a set of real data on widths of shrubs.

Novel empirical likelihood inference for the mean difference with right-censored data.

This paper focuses on comparing two means and finding a confidence interval for the difference of two means with right-censored data using the empirical likelihood method combined with the independent and identically distributed random functions representation. In the literature, some early researchers proposed empirical link-based confidence intervals for the mean difference based on right-censored data using the synthetic data approach. However, their empirical log-likelihood ratio statistic has a scaled chi-squared distribution. To avoid the estimation of the scale parameter in constructing confidence intervals, we propose an empirical likelihood method based on the independent and identically distributed representation of Kaplan-Meier weights involved in the empirical likelihood ratio. We obtain the standard chi-squared distribution. We also apply the adjusted empirical likelihood to improve coverage accuracy for small samples. In addition, we investigate a new empirical likelihood method, the mean empirical likelihood, within the framework of our study. The performances of all the empirical likelihood methods are compared via extensive simulations. The proposed empirical likelihood-based confidence interval has better coverage accuracy than those from existing methods. Finally, our findings are illustrated with a real data set.

Statistical Inference for the Heteroscedastic Partially Linear Varying-Coefficient Errors-in-Variables Model with Missing Censoring Indicators

In this paper, we focus on heteroscedastic partially linear varying-coefficient errors-in-variables models under right-censored data with censoring indicators missing at random. Based on regression calibration, imputation, and inverse probability weighted methods, we define a class of modified profile least square estimators of the parameter and local linear estimators of the coefficient function, which are applied to constructing estimators of the error variance function. In order to improve the estimation accuracy and take into account the heteroscedastic error, reweighted estimators of the parameter and coefficient function are developed. At the same time, we apply the empirical likelihood method to construct confidence regions and maximum empirical likelihood estimators of the parameter. Under appropriate assumptions, the asymptotic normality of the proposed estimators is studied. The strong uniform convergence rate for the estimators of the error variance function is considered. Also, the asymptotic chi-squared distribution of the empirical log-likelihood ratio statistics is proved. A simulation study is conducted to evaluate the finite sample performance of the proposed estimators. Meanwhile, one real data example is provided to illustrate our methods.

Orthogonality-based empirical likelihood inference for varying-coefficient partially nonlinear model with longitudinal data

In this paper, we study empirical likelihood-based inference for longitudinal data with varying-coefficient partially nonlinear model. Based on the orthogonality estimation technology, the QR decomposition is firstly used to separate the nonparametric component in the model. With the quadratic inference functions (QIF), we propose an estimator for the parameter that avoids estimating the nuisance parameter in the correlation matrix directly. In addition, we construct an empirical log-likelihood ratio statistic for the parameter and obtain the maximum empirical likelihood (MEL) estimator. The proposed MEL estimator has the same asymptotic variance as the QIF estimator and is more efficient than the estimator from the conventional generalized estimating equations (GEE). Under some assumptions, we establish certain asymptotic properties of the resulting estimators. Furthermore, we conduct simulation studies to evaluate the performances of the proposed estimation procedures in finite samples.

Influence function-based empirical likelihood for inference of quantile medical costs with censored data.

In this paper, we propose empirical likelihood methods based on influence function and Jackknife techniques to construct confidence intervals for quantile medical costs with censored data. We show that the influence function-based empirical log-likelihood ratio statistic for the quantile medical cost has a standard Chi-square distribution as its asymptotic distribution. Simulation studies are conducted to compare coverage probabilities and interval lengths of the proposed empirical likelihood confidence intervals with the existing normal approximation-based confidence intervals for quantile medical costs. The proposed methods are observed to have better finite-sample performances than existing methods. The new methods are also illustrated through a real example.

Empirical likelihood for high-dimensional partially functional linear model

This paper considers empirical likelihood inference for a high-dimensional partially functional linear model. An empirical log-likelihood ratio statistic is constructed for the regression coefficients of non-functional predictors and proved to be asymptotically normally distributed under some regularity conditions. Moreover, maximum empirical likelihood estimators of the regression coefficients of non-functional predictors are proposed and their asymptotic properties are obtained. Simulation studies are conducted to demonstrate the performance of the proposed procedure and a real data set is analyzed for illustration.

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

Abstract In this paper, we investigate a partially single-index varying-coefficient model, and suggest two empirical log-likelihood ratio statistics for the unknown parameters in the model. The first statistic is asymptotically distributed as a weighted sum of independent chi-square variables under some mild conditions. It is proved that another statistic, with adjustment factor, is asymptotically standard chi-square under some suitable conditions. These useful statistics could be used to construct the confidence regions of the parameters. A simulation study indicates that, with the increase of sample size, the coverage probability of the confidence region constructed by us gradually approaches the theoretical value.

Influence function-based empirical likelihood and generalized confidence intervals for the Lorenz curve

This paper aims to solve confidence interval estimation problems for the Lorenz curve. First, we propose new nonparametric confidence intervals using the influence function-based empirical likelihood method. We show that the limiting distributions of the empirical log-likelihood ratio statistics for the Lorenz ordinates are standard chi-square distributions. We also develop “exact” parametric intervals for the Lorenz ordinate based on generalized pivotal quantities when the underlying income distribution is a Pareto distribution or a Lognormal distribution. Extensive simulation studies are conducted to evaluate the finite sample performances of the proposed methods. Finally, we apply our methods to a real income dataset.

Statistical inference for varying-coefficient partially linear errors-in-variables models with missing data

The purpose of this paper is twofold. First, we investigate estimations in varying-coefficient partially linear errors-in-variables models with covariates missing at random. However, the estimators are often biased due to the existence of measurement errors, the bias-corrected profile least-squares estimator and local liner estimators for unknown parametric and coefficient functions are obtained based on inverse probability weighted method. The asymptotic properties of the proposed estimators both for the parameter and nonparametric parts are established. Second, we study asymptotic distributions of an empirical log-likelihood ratio statistic and maximum empirical likelihood estimator for the unknown parameter. Based on this, more accurate confidence regions of the unknown parameter can be constructed. The methods are examined through simulation studies and illustrated by a real data analysis.

Empirical likelihood for first-order mixed integer-valued autoregressive model

In this paper, we not only construct the confidence region for parameters in a mixed integer-valued autoregressive process using the empirical likelihood method, but also establish the empirical log-likelihood ratio statistic and obtain its limiting distribution. And then, via simulation studies we give coverage probabilities for the parameters of interest. The results show that the empirical likelihood method performs very well.

An empirical likelihood approach under cluster sampling with missing observations

The parameter of interest considered is the unique solution to a set of estimating equations, such as regression parameters of generalised linear models. We consider a design-based approach; that is, the sampling distribution is specified by stratification, cluster (multi-stage) sampling, unequal selection probabilities, side information and a response mechanism. The proposed empirical likelihood approach takes into account of these features. Empirical likelihood has been mostly developed under more restrictive settings, such as independent and identically distributed assumption, which is violated under a design-based framework. A proper empirical likelihood approach which deals with cluster sampling, missing data and multidimensional parameters is absent in the literature. This paper shows that a cluster-level empirical log-likelihood ratio statistic is pivotal. The main contribution of the paper is to provide the rigorous asymptotic theory and underlining regularity conditions which imply $${\surd {n}}$$-consistency and the Wilks’s theorem or self-normalisation property. Negligible and large sampling fractions are considered.

Empirical likelihood for heteroscedastic partially linear single-index models with growing dimensional data

In this paper, we propose a new approach to the empirical likelihood inference for the parameters in heteroscedastic partially linear single-index models. In the growing dimensional setting, it is proved that estimators based on semiparametric efficient score have the asymptotic consistency, and the limit distribution of the empirical log-likelihood ratio statistic for parameters $$(\beta ^{\top },\theta ^{\top })^{\top }$$ is a normal distribution. Furthermore, we show that the empirical log-likelihood ratio based on the subvector of $$\beta $$ is an asymptotic chi-square random variable, which can be used to construct the confidence interval or region for the subvector of $$\beta $$ . The proposed method can naturally be applied to deal with pure single-index models and partially linear models with high-dimensional data. The performance of the proposed method is illustrated via a real data application and numerical simulations.

Jackknife empirical likelihood for the difference of two volumes under ROC surfaces

The volume under a surface (VUS) is an effective measure for evaluating the discriminating power of a diagnostic test with three ordinal diagnostic groups. In this paper, we investigate the difference of two correlated VUS’s to compare two treatments for discrimination of three-class classification data. A jackknife empirical likelihood (JEL) procedure is employed to avoid the variance estimation in the existing methods. We prove that the limiting distribution of the empirical log-likelihood ratio statistic follows a $$\chi ^2$$ distribution. Extensive numerical studies show that the JEL confidence intervals outperform those based on the normal approximation method. The proposed method is also applied to the Alzheimer’s disease data.

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

ABSTRACTIn 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 high-dimensional partially linear model with martingale difference errors

ABSTRACTIn 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.

Jackknife empirical likelihood confidence regions for the evaluation of continuous-scale diagnostic tests with verification bias

Recently, Wang and Qin proposed various bias-corrected empirical likelihood confidence regions for any two of the three parameters, sensitivity, specificity, and cut-off value, with the remaining parameter fixed at a given value in the evaluation of a continuous-scale diagnostic test with verification bias. In order to apply those methods, quantiles of the limiting weighted chi-squared distributions of the empirical log-likelihood ratio statistics should be estimated. In order to facilitate application and reduce computation burden, in this paper, jackknife empirical likelihood-based methods are proposed for any pairs of sensitivity, specificity and cut-off value, and asymptotic results can be derived accordingly. The proposed methods can be easily implemented to construct confidence regions for the evaluation of continuous-scale diagnostic tests with verification bias. Simulation studies are conducted to evaluate the finite sample performance and robustness of the proposed jackknife empirical likelihood-based confidence regions in terms of coverage probabilities. Finally, a real case analysis is provided to illustrate the application of new methods.

Jackknife empirical likelihood inference for the mean absolute deviation

In statistics mean absolute deviation plays an important role in measuring spread of a data. In this paper, we focus on using the jackknife, the adjusted and the extended jackknife empirical likelihood methods to construct confidence intervals for the mean absolute deviation of a random variable. The empirical log-likelihood ratio statistic is derived whose asymptotic distribution is a standard chi-square distribution. The results of simulation study show the comparison of the average length and coverage probability by using jackknife empirical likelihood methods and normal approximation method. The proposed adjusted and extended jackknife empirical likelihood methods perform better than other methods, in particular for skewed distributions. We use real data sets to illustrate the proposed jackknife empirical likelihood methods.

Empirical likelihood for semivarying coefficient model with measurement error in the nonparametric part

A semivarying coefficient model with measurement error in the nonparametric part was proposed by Feng and Xue (Ann Inst Stat Math 66:121–140, 2014), but its inferences have not been systematically studied. This paper applies empirical likelihood method to construct confidence regions/intervals for the regression parameter and coefficient function. When some auxiliary information about the parametric part is available, the empirical log-likelihood ratio statistic for the regression parameter is introduced based on the corrected local linear estimator of the coefficient function. Furthermore, corrected empirical log-likelihood ratio statistic for coefficient function is also investigated with the use of auxiliary information. The limiting distributions of the resulting statistics both for the regression parameter and coefficient function are shown to have standard Chi-squared distribution. Simulation experiments and a real data set are presented to evaluate the finite sample performance of our proposed method.

Empirical likelihood for linear and log-linear INGARCH models

The integer-valued GARCH model is a popular tool for modeling time series of counts. This paper develops empirical likelihood methods for the linear and log-linear integer-valued GARCH models, respectively. We consider three aspects of the empirical likelihood method: estimates, confidence regions and test. For both models, an empirical log-likelihood ratio statistic is derived and its asymptotic distribution is shown to be a chi-squared distribution. Simulation studies lead to superior performance of the empirical likelihood method compared with the conventional treatments in the literature. An application to a real data example is also provided.