Empirical Likelihood for Semiparametric Varying-Coefficient Heteroscedastic Partially Linear Errors-in-Variables Models

The purpose of this article is to use the empirical likelihood method to study construction of the confidence region for the parameter of interest in semiparametric varying-coefficient heteroscedastic partially linear errors-in-variables models. When the variance functions of the errors are known or unknown, we propose the empirical log-likelihood ratio statistics for the parameter of interest. For each case, a nonparametric version of Wilks’ theorem is derived. The results are then used to construct confidence regions of the parameter. A simulation study is carried out to assess the performance of the empirical likelihood method.     

Empirical Likelihood for Partial Linear Models under Negatively Associated Errors

In this paper, we use blockwise empirical likelihood (EL) technique to construct confidence regions for the parameter of the partial linear models under negatively associated errors. It is shown that the blockwise EL ratio statistic for the parameter of interest is asymptotically χ²-type distributed by employing the large-block and small-block arguments. The result can be used to obtain confidence regions for the parameter of interest.     

Empirical Likelihood for Single-Index Regression Models under Negatively Associated Errors

In this article, we use bockwise empirical likelihood technique to construct confidence regions for the parameter of the single-index models under negatively associated errors. It is shown that the blockwise empirical likelihood ratio statistic for the parameter of interest is asymptotically χ²-type distributed. The result can be used to obtain confidence regions for the parameter of interest.     

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 for generalized partially linear varying-coefficient models

Generalized partially linear varying-coefficient models (GPLVCM) are frequently used in statistical modeling. However, the statistical inference of the GPLVCM, such as confidence region/interval construction, has not been very well developed. In this article, empirical likelihood-based inference for the parametric components in the GPLVCM is investigated. Based on the local linear estimators of the GPLVCM, an estimated empirical likelihood-based statistic is proposed. We show that the resulting statistic is asymptotically non-standard chi-squared. By the proposed empirical likelihood method, the confidence regions for the parametric components are constructed. In addition, when some components of the parameter are of particular interest, the construction of their confidence intervals is also considered. A simulation study is undertaken to compare the empirical likelihood and the other existing methods in terms of coverage accuracies and average lengths. The proposed method is applied to a real example.     

Empirical likelihood for heteroscedastic partially linear errors-in-variables model with α-mixing errors

In this paper, we apply the empirical likelihood method to heteroscedastic partially linear errors-in-variables model. For the cases of known and unknown error variances, the two different empirical log-likelihood ratios for the parameter of interest are constructed. If the error variances are known, the empirical log-likelihood ratio is proved to be asymptotic chi-square distribution under the assumption that the errors are given by a sequence of stationary α-mixing random variables. Furthermore, if the error variances are unknown, we show that the proposed statistic is asymptotically standard chi-square distribution when the errors are independent. Simulations are carried out to assess the performance of the proposed method.     

Empirical Likelihood Inference for the Parameter in Additive Partially Linear EV Models

In this article, we consider empirical likelihood inference for the parameter in the additive partially linear models when the linear covariate is measured with error. By correcting for attenuation, a corrected-attenuation empirical log-likelihood ratio statistic for the unknown parameter β, which is of primary interest, is suggested. We show that the proposed statistic is asymptotically standard chi-square distribution without requiring the undersmoothing of the nonparametric components, and hence it can be directly used to construct the confidence region for the parameter β. Some simulations indicate that, in terms of comparison between coverage probabilities and average lengths of the confidence intervals, the proposed method performs better than the profile-based least-squares method. We also give the maximum empirical likelihood estimator (MELE) for the unknown parameter β, and prove the MELE is asymptotically normal under some mild conditions.     

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

Abstract

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 inference for semiparametric model with linear process errors

The purpose of this article is to use the empirical likelihood method to study the confidence regions construction for the parameters of interest in semiparametric model with linear process errors under martingale difference. It is shown that the adjusted empirical log-likelihood ratio at the true parameters is asymptotically chi-squared. A simulation study indicates that the adjusted empirical likelihood works better than a normal approximation-based approach.     

The Empirical Likelihood for First-Order Random Coefficient Integer-Valued Autoregressive Processes

This article studies the empirical likelihood method for the first-order random coefficient integer-valued autoregressive process. The limiting distribution of the log empirical likelihood ratio statistic is established. Confidence region for the parameter of interest and its coverage probabilities are given, and hypothesis testing is considered. The maximum empirical likelihood estimator for the parameter is derived and its asymptotic properties are established. The performances of the estimator are compared with the conditional least squares estimator via simulation.     

Empirical Likelihood for Partially Linear Single-Index Errors-in-Variables Model

In this article, we consider the application of the empirical likelihood method to a partially linear single-index model. We focus on the case where some covariates are measured with additive errors. It is shown that the empirical log-likelihood ratio at the true parameter converges to the standard chi-square distribution. Simulations show that the proposed confidence region has coverage probability which is closer to the nominal level, as well as narrower than those of normal approximation method. A real data example is given.     

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.     

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 dimension reduction inference for partially non-linear error-in-responses models with validation data

ABSTRACT

In this article, partially non linear models when the response variable is measured with error and explanatory variables are measured exactly are considered. Without specifying any error structure equation, a semiparametric dimension reduction technique is employed. Two estimators of unknown parameter in non linear function are obtained and asymptotic normality is proved. In addition, empirical likelihood method for parameter vector is provided. It is shown that the estimated empirical log-likelihood ratio has asymptotic Chi-square distribution. A simulation study indicates that, compared with normal approximation method, empirical likelihood method performs better in terms of coverage probabilities and average length of the confidence intervals.     

Quasi-likelihood from<Emphasis Type="Italic">M</Emphasis>-estimators: A numerical comparison with empirical likelihood

In this paper we compare two robust pseudo-likelihoods for a parameter of interest, also in the presence of nuisance parameters. These functions are obtained by computing quasi-likelihood and empirical likelihood from the estimating equations which define robustM-estimators. Application examples in the context of linear transformation models are considered. Monte Carlo studies are performed in order to assess the finite-sample performance of the inferential procedures based on quasi-and empirical likelihood, when the objective is the construction of robust confidence regions.     

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.     

Numerical Comparison Between Different Empirical Prediction Intervals Under the Fay-Herriot Model

Recently, an empirical best linear unbiased predictor is widely used as a practical approach to small area inference. It is also of interest to construct empirical prediction intervals. However, we do not know which method should be used from among the several existing prediction intervals. In this article, we first obtain an empirical prediction interval by using the residual maximum likelihood method for estimating unknown model variance parameters. Then we compare the later with other intervals with the residual maximum likelihood method. Additionally, some different parametric bootstrap methods for constructing empirical prediction intervals are also compared in a simulation study.     

缺失数据下广义线性模型的经验似然推断

利用经验似然方法,讨论缺失数据下广义线性模型中参数的置信域问题,得到了对数经验似然比统计量的渐近分布为标准卡方分布;给出参数的一些估计量及其渐近分布,利用数据模拟解释了所提出的方法。     

Empirical Likelihood-based Inference in Linear Models with Missing Data

The missing response problem in linear regression is studied. An adjusted empirical likelihood approach to inference on the mean of the response variable is developed. A non-parametric version of Wilks's theorem for the adjusted empirical likelihood is proved, and the corresponding empirical likelihood confidence interval for the mean is constructed. With auxiliary information, an empirical likelihood-based estimator with asymptotic normality is defined and an adjusted empirical log-likelihood function with asymptotic χ² is derived. A simulation study is conducted to compare the adjusted empirical likelihood methods and the normal approximation methods in terms of coverage accuracies and average lengths of the confidence intervals. Based on biases and standard errors, a comparison is also made between the empirical likelihood-based estimator and related estimators by simulation. Our simulation indicates that the adjusted empirical likelihood methods perform competitively and the use of auxiliary information provides improved inferences.     

Empirical likelihood inference for parameters in a partially linear errors-in-variables model

In this paper, we consider the application of the empirical likelihood method to a partially linear model with measurement errors in the non-parametric part. It is shown that the empirical log-likelihood ratio at the true parameters converges to the standard chi-square distribution. Furthermore, we obtain the maximum empirical likelihood estimate of the unknown parameter by using the empirical log-likelihood ratio function, and the resulting estimator is shown to be asymptotically normal. Some simulations and an application are conducted to illustrate the proposed method.