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 Confidence Intervals for Distribution Functions under Negatively Associated Samples

In this article, we discuss the construction of the confidence intervals for distribution functions under negatively associated samples. It is shown that the blockwise empirical likelihood (EL) ratio statistic for a distribution function is asymptotically χ²-type distributed. The result is used to obtain an EL-based confidence interval for the distribution function.     

Quantile Estimation in the Presence of Auxiliary Information under Negatively Associated Samples

In this article, we apply the empirical likelihood technique to propose a new class of quantile estimators in the presence of some auxiliary information under negatively associated samples. It is shown that the proposed quantile estimators are asymptotically normally distributed with smaller asymptotic variances than those of the usual quantile estimators. It is also shown that blocking technique is an useful tool in estimating asymptotic variance under negatively associated samples, which makes it possible to construct normal approximation based confidence intervals for quantiles.     

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.     

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 probability density functions under negatively associated samples

In this paper, we study the construction of confidence intervals for a probability density function under a negatively associated sample 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 interval on the probability density function.     

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.     

Empirical Likelihood Method for Censored Median Regression Models

Based on the semiparametric median regression analysis for the right-censored data developed by Ying et al. (1995 Ying , Z. , Jung , S. H. , Wei , L. J. ( 1995 ). Survival analysis with median regression models . J. Amer. Statist. Assoc. 90 : 178 – 184 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]), an empirical likelihood based inferential procedure for the regression coefficients is proposed. The limiting distribution of the proposed log-empirical likelihood ratio test statistic follows a chi-squared distribution, which corresponds to the standard asymptotic results of the empirical likelihood method. The inference about the subsets of the entire regression coefficients vector is discussed. The proposed method is illustrated by some simulation studies.     

Empirical Likelihood for Nonparametric Components in Additive Partially Linear Models

Empirical likelihood-based inference for the nonparametric components in additive partially linear models is investigated. An empirical likelihood approach to construct the confidence intervals of the nonparametric components is proposed when the linear covariate is measured with and without errors. We show that the proposed empirical log-likelihood ratio is asymptotically standard chi-squared without requiring the undersmoothing of the nonparametric components. Then, it can be directly used to construct the confidence intervals for the nonparametric functions. A simulation study indicates that, compared with a normal approximation-based approach, the proposed method works better in terms of coverage probabilities and widths of the pointwise confidence intervals.     

A Goodness-of-Fit Test for Logistic Regression Models in Stratified Case-Control Studies via Empirical Likelihood

In the literature, there were only a few reports on goodness-of-fit tests on logistic regression models specifically derived for case-control studies. In this article, we propose a goodness-of-fit test for logistic regression models in stratified case-control studies using an empirical likelihood approach. The proposed statistic is an alternative to the statistic G _o , recently proposed by Arbigast and Lin (2005 Arbigast , P. G. , Lin , D. Y. ( 2005 ). Model-checking techniques for stratified case-control studies . Statist. Med. 24 : 229 – 247 . [Google Scholar]). Simulation results show that the proposed statistic is often slightly more powerful than G _o , although their performances are always close to each other. Moreover, implementation of our method is easy since the usual stratified logistic regression procedures in many statistical softwares can be employed. Some asymptotic results and application of the proposed statistic to two real datasets are also presented.     

Empirical Likelihood Ratio for Linear Transformation Models with Doubly Censored Data

Double censoring arises when T represents an outcome variable that can only be accurately measured within a certain range, [L, U], where L and U are the left- and right-censoring variables, respectively. When L is always observed, we consider the empirical likelihood inference for linear transformation models, based on the martingale-type estimating equation proposed by Chen et al. (2002 Chen , K. , Jin , Z. , Ying , Z. ( 2002 ). Semiparametric analysis of transformation models with censored data . Biometrika 89 : 659 – 668 .[Crossref], [Web of Science ®] , [Google Scholar]). It is demonstrated that both the approach of Lu and Liang (2006 Lu , W. , Liang , Y. ( 2006 ). Empirical likelihood inference for linear transformation models . Journal of Multivariate Analysis 97 : 1586 – 1599 .[Crossref], [Web of Science ®] , [Google Scholar]) and that of Yu et al. (2011 Yu , W. , Sun , Y. , Zheng , M. ( 2011 ). Empirical likelihood method for linear transformation models . Annals of the Institute of Statistical Mathematics 63 : 331 – 346 .[Crossref], [Web of Science ®] , [Google Scholar]) can be extended to doubly censored data. Simulation studies are conducted to investigate the performance of the empirical likelihood ratio methods.     

Empirical Likelihood for Threshold Autoregressive Models

This article develops empirical likelihood for threshold autoregressive models. We propose general estimating equations based on moment constraint. Under some suitable conditions, we show the empirical likelihood estimators for parameter are asymptotically normally distributed, and the proposed log empirical likelihood ratio statistic asymptotically follows a standard chi-squared distribution.     

Empirical Likelihood Ratio Test for a Change-Point in Linear Regression Model

A nonparametric method based on the empirical likelihood is proposed to detect the change-point in the coefficient of linear regression models. The empirical likelihood ratio test statistic is proved to have the same asymptotic null distribution as that with classical parametric likelihood. Under some mild conditions, the maximum empirical likelihood change-point estimator is also shown to be consistent. The simulation results show the sensitivity and robustness of the proposed approach. The method is applied to some real datasets to illustrate the effectiveness.     

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.     

Uniform Test for Predictive Regression With AR Errors

Testing predictability is of importance in economics and finance. Based on a predictive regression model with independent and identically distributed errors, some uniform tests have been proposed in the literature without distinguishing whether the predicting variable is stationary or nearly integrated. In this article, we extend the empirical likelihood methods of Zhu, Cai, and Peng with independent errors to the case of an AR error process. Again, the proposed new tests do not need to know whether the predicting variable is stationary or nearly integrated, and whether it has a finite variance or an infinite variance. A simulation study shows the new methodologies perform well in finite sample.     

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.     

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 Nonparametric Models Under Linear Process Errors

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

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.     

Empirical Likelihood for First-order Autoregressive Error-in-variable of Models With Validation Data

In this article, we consider the empirical likelihood for the autoregressive error-in-explanatory variable models. With the help of validation, we first develop an empirical likelihood ratio test statistic for the parameters of interest, and prove that its asymptotic distribution is that of a weighted sum of independent standard χ²₁ random variables with unknown weights. Also, we propose an adjusted empirical likelihood and prove that its asymptotic distribution is a standard χ². Furthermore, an empirical likelihood-based confidence region is given. Simulation results indicate that the proposed method works well for practical situations.