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.     

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.     

Bartlett-corrected two-sample adjusted empirical likelihood via resampling

To construct confidence regions for the difference of two population means, Liu and Yu (2010 Liu, Y., Yu, C.W. (2010). Bartlett correctable two-sample adjusted empirical likelihood. J. Multivariate Anal. 101(7):1701–1711.[Crossref], [Web of Science ®] , [Google Scholar]) proposed a two-sample adjusted empirical likelihood (AEL) with high-order precision. However, two issues have not been well addressed. The first one is that the AEL ratio function is bounded such that the size of the confidence regions may overly expand when the sample sizes are small and/or the dimension of data is large. The second issue is that its high-order precision relies on accurate estimation of the Bartlett factor, while accurately estimating the Bartlett factor is a serious challenge. In order to address these two problems simultaneously, we propose a two-sample modified AEL to ensure the boundedness of confidence regions and preserve the Bartlett correctability. A two-stage procedure is proposed for constructing accurate confidence regions via resampling. The finite-sample performance of the proposed method is illustrated by simulations and a real-data example.     

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.     

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.     

Jackknife empirical likelihood for the error variance in linear models

Variance estimation is a fundamental yet important problem in statistical modelling. In this paper, we propose jackknife empirical likelihood (JEL) methods for the error variance in a linear regression model. We prove that the JEL ratio converges to the standard chi-squared distribution. The asymptotic chi-squared properties for the adjusted JEL and extended JEL estimators are also established. Extensive simulation studies to compare the new JEL methods with the standard method in terms of coverage probability and interval length are conducted, and the simulation results show that our proposed JEL methods perform better than the standard method. We also illustrate the proposed methods using two real data sets.     

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.     

A note on the two-sample mean problem based on jackknife empirical likelihood

In this article, we employ the jackknife empirical likelihood (JEL) method to construct the confidence regions for the difference of the means of two d-dimensional samples. Compared with traditional EL for the two-sample mean problem, JEL is extremely simpler to use in practice and is more effective in computing. Based on the JEL ratio test, a version of Wilks’ theorem is developed. Furthermore, to improve the coverage accuracy of confidence regions, a Bartlett correction is applied. The effectiveness of the proposed method is demonstrated by a simulation study and a real data analysis.     

Adjusted empirical likelihood for value at risk and expected shortfall

Value at risk (VaR) and expected shortfall (ES) are widely used risk measures of the risk of loss on a specific portfolio of financial assets. Adjusted empirical likelihood (AEL) is an important non parametric likelihood method which is developed from empirical likelihood (EL). It can overcome the limitation of convex hull problems in EL. In this paper, we use AEL method to estimate confidence region for VaR and ES. Theoretically, we find that AEL has the same large sample statistical properties as EL, and guarantees solution to the estimating equations in EL. In addition, simulation results indicate that the coverage probabilities of the new confidence regions are higher than that of the original EL with the same level. These results show that the AEL estimation for VaR and ES deserves to recommend for the real applications.     

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.     

Block empirical likelihood for longitudinal partially linear regression models

The authors propose a block empirical likelihood procedure to accommodate the within‐group correlation in longitudinal partially linear regression models. This leads them to prove a nonparametric version of the Wilks theorem. In comparison with normal approximations, their method does not require a consistent estimator for the asymptotic covariance matrix, which makes it easier to conduct inference on the parametric component of the model. An application to a longitudinal study on fluctuations of progesterone level in a menstrual cycle is used to illustrate the procedure developed here.     

Smoothed empirical likelihood for GARCH models with heavy-tailed errors

ABSTRACT

This paper proposes an empirical likelihood (EL) method for estimating the GARCH(p, q) models with heavy-tailed errors. Using the kernel smoothing method, we derive a smoothed EL ratio statistic, which yields a smoothed EL estimator. Moreover, we derive a profile EL for the partial parameters in the presence of nuisance parameters. Simulations and empirical results are conducted to illustrate our proposed method.     

Comparison of empirical likelihood and its dual likelihood under density ratio model

Density ratio models (DRMs) are commonly used semiparametric models to link related populations. Empirical likelihood (EL) under DRM has been demonstrated to be a flexible and useful platform for semiparametric inferences. Since DRM-based EL has the same maximum point and maximum likelihood as its dual form (dual EL), EL-based inferences under DRM are usually made through the latter. A natural question comes up: is there any efficiency loss of doing so? We make a careful comparison of the dual EL and DRM-based EL estimation methods from theory and numerical simulations. We find that their point estimators for any parameter are exactly the same, while they may have different performances in interval estimation. In terms of coverage accuracy, the two intervals are comparable for non- or moderate skewed populations, and the DRM-based EL interval can be much superior for severely skewed populations. A real data example is analysed for illustration purpose.     

Robust empirical likelihood for linear models under median constraints

In this paper, we consider the application of the empirical likelihood for

linear models under median constraints in view of robustness. For two simple median constraints, it is shown that conditions to ensure the consistency of the empirical likelihood confidence regions can be surprisingly relaxed compared with the normal approach under L norm. However, the coverage accuracy of the empirical likelihood confidence regions based on simple median constrains cannot be improved because of the discontinuity of the constraints. Therefore, a smoothed version of median constraint is proposed and a general theory is established to ensure its validity.     

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.     

Semiparametric test for multiple change-points based on empirical likelihood

In the dynamic financial market, the change of financial asset prices is always described as a certain random events which result in abrupt changes. The random time when the event occurs is called a change point. As the event happens, in order to mitigate property damage the government should increase the macro-control ability. As a result, we need to find a valid statistical model for change point problem to solve it effectively. This paper proposes a semiparametric model for detecting the change points. According to the research of empirical studies and hypothesis testing we acquire the maximum likelihood estimators of change points. We use the loglikelihood ratio to test the multiple change points. We obtain some asymptotic results. The estimated change point is more efficient than the non parametric one through simulation experiments. Real data application illustrates the usage of the model.     

An extensive power evaluation of a novel two-sample density-based empirical likelihood ratio test for paired data with an application to a treatment study of attention-deficit/hyperactivity disorder and severe mood dysregulation

In many case-control studies, it is common to utilize paired data when treatments are being evaluated. In this article, we propose and examine an efficient distribution-free test to compare two independent samples, where each is based on paired observations. We extend and modify the density-based empirical likelihood ratio test presented by Gurevich and Vexler [7] to formulate an appropriate parametric likelihood ratio test statistic corresponding to the hypothesis of our interest and then to approximate the test statistic nonparametrically. We conduct an extensive Monte Carlo study to evaluate the proposed test. The results of the performed simulation study demonstrate the robustness of the proposed test with respect to values of test parameters. Furthermore, an extensive power analysis via Monte Carlo simulations confirms that the proposed method outperforms the classical and general procedures in most cases related to a wide class of alternatives. An application to a real paired data study illustrates that the proposed test can be efficiently implemented in practice.