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.     

Adjusted empirical likelihood models with estimating equations for accelerated life tests

This article proposes an adjusted empirical likelihood estimation (AMELE) method to model and analyze accelerated life testing data. This approach flexibly and rigorously incorporates distribution assumptions and regression structures by estimating equations within a semiparametric estimation framework. An efficient method is provided to compute the empirical likelihood estimates, and asymptotic properties are studied. Real-life examples and numerical studies demonstrate the advantage of the proposed methodology.     

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.     

Performance of and software for a modified mantel-haenszel statistic for correlated data

This paper presents the results of a small sample simulation study designed to evaluate the performance of a recently proposed test statistic for the analysis of correlated binary data. The new statistic is an adjusted Mantel-Haenszel test, which may be used in testing for association between a binary exposure and a binary outcome of interest across several fourfold tables when the data have been collected under a cluster sampling design. Al- though originally developed for the analysis of periodontal data, the proposed method may be applied to clustered binary data arising in a variety of settings, including longitu- dinal studies, family studies, and school-based research. The features of the simulation are intended to mimic those of a research study of periodontal health, in which a large number of observations is made on each of a relatively small number of patients. The simulation reveals that the adjusted test statistic performs well in finite samples, having empirical type I error rates close to nominal and empirical power similar to that of more complicated marginal regression methods. Software for computing the adjusted statistic is also provided.     

Empirical likelihood for linear regression models under imputation for missing responses

The authors study the empirical likelihood method for linear regression models. They show that when missing responses are imputed using least squares predictors, the empirical log‐likelihood ratio is asymptotically a weighted sum of chi‐square variables with unknown weights. They obtain an adjusted empirical log‐likelihood ratio which is asymptotically standard chi‐square and hence can be used to construct confidence regions. They also obtain a bootstrap empirical log‐likelihood ratio and use its distribution to approximate that of the empirical log‐likelihood ratio. A simulation study indicates that the proposed methods are comparable in terms of coverage probabilities and average lengths of confidence intervals, and perform better than a normal approximation based method.     

Adjusted empirical likelihood for right censored lifetime data

Adjusted empirical likelihood (AEL) is a method to improve the performance of the empirical likelihood (EL) particularly in the construction of the confidence interval based on completely observed data. In this paper, we extend AEL approach to the analysis of right censored data by adopting an influence function method. The main results include that the adjusted log-likelihood ratio is asymptotically Chi-squared distributed. Simulation results indicate that the proposed AEL-based confidence intervals perform better compared with normality-based or EL-based confidence intervals specifically for small sample size within the right-censoring setting. The proposed method is illustrated by analysis of survival time of patients after operation for spinal tumors.     

Two Adjusted Empirical-Likelihood-Based Methods in Generalized Varying-Coefficient Partially Linear Model

An empirical likelihood method was proposed by Owen and has been extended to many semiparametric and nonparametric models with a continuous response variable. However, there has been less attention focused on the generalized regression model. This article systematically studies two adjusted empirical-likelihood-based methods in the generalized varying-coefficient partially linear models. Based on the popular profile likelihood estimation procedure, the new adjusted empirical likelihood technology for the parameter is established and the resulting statistics are shown to be asymptotically standard chi-square distributed. Further, the adjusted empirical-likelihood-based confidence regions are established, and an efficient adjusted profile empirical-likelihood-based confidence intervals/regions for any components of the parameter, which are of primary interest, is also constructed. Their asymptotic properties are also derived. Some numerical studies are carried out to illustrate the performance of the proposed inference procedures.     

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.     

Profile empirical-likelihood inferences for the single-index-coefficient regression model

This article deals with a new profile empirical-likelihood inference for a class of frequently used single-index-coefficient regression models (SICRM), which were proposed by Xia and Li (J. Am. Stat. Assoc. 94:1275–1285, 1999a). Applying the empirical likelihood method (Owen in Biometrika 75:237–249, 1988), a new estimated empirical log-likelihood ratio statistic for the index parameter of the SICRM is proposed. To increase the accuracy of the confidence region, a new profile empirical likelihood for each component of the relevant parameter is obtained by using maximum empirical likelihood estimators (MELE) based on a new and simple estimating equation for the parameters in the SICRM. Hence, the empirical likelihood confidence interval for each component is investigated. Furthermore, corrected empirical likelihoods for functional components are also considered. The resulting statistics are shown to be asymptotically standard chi-squared distributed. Simulation studies are undertaken to assess the finite sample performance of our method. A study of real data is also reported.     

Modified tests for change points in variance in the possible presence of mean breaks

This paper considers the detection problem of variance changes for the time series involving abrupt and/or smooth breaks in mean. Often, in these situations, the tests of choice are based on cumulative sum of squares statistics. We show that the test statistics are not robust in the presence of broken mean and their sizes suffer severe distortions. The adjusted residual-based method is then proposed to eliminate these deficiencies and makes a significant improvement. Finally, simulation results confirm the validity of these modified test statistics, and an empirical data analysis using some stock price series from the Shanghai Stock Exchange is reported.     

Functional Partial Linear Single‐index Model

This paper deals with the problem of predicting the real‐valued response variable using explanatory variables containing both multivariate random variable and random curve. The proposed functional partial linear single‐index model treats the multivariate random variable as linear part and the random curve as functional single‐index part, respectively. To estimate the non‐parametric link function, the functional single‐index and the parameters in the linear part, a two‐stage estimation procedure is proposed. Compared with existing semi‐parametric methods, the proposed approach requires no initial estimation and iteration. Asymptotical properties are established for both the parameters in the linear part and the functional single‐index. The convergence rate for the non‐parametric link function is also given. In addition, asymptotical normality of the error variance is obtained that facilitates the construction of confidence region and hypothesis testing for the unknown parameter. Numerical experiments including simulation studies and a real‐data analysis are conducted to evaluate the empirical performance of the proposed method.     

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.     

基于多元经验模式分解的股票收益与宏观经济关系分析

提出一种基于多元经验模式分解的股票市场收益与宏观经济活动关系的分析方法。通过月度道琼斯指数和美国工业生产指数的联合多元经验模式分解,得到多元金融时间序列的多尺度分量。采用希尔伯特—黄变换和边际谱确定每个尺度的主周期,进而在不同尺度下对多元时间序列进行相关性分析及Granger因果检验。结果表明:股票指数在中、长周期的某些尺度上是工业生产指数的Granger原因,序列之间具有明显的相关性,股票指数领先工业生产指数16个月到32个月不等。     

Empirical likelihood for density-weighted average derivatives

In this paper, we investigate empirical likelihood (EL) inference for density-weighted average derivatives in nonparametric multiple regression models. A simply adjusted empirical log-likelihood ratio for the vector of density-weighted average derivatives is defined and its limiting distribution is shown to be a standard Chi-square distribution. To increase the accuracy and coverage probability of confidence regions, an EL inference procedure for the rescaled parameter vector is proposed by using a linear instrumental variables regression. The new method shares the same properties of the regular EL method with i.i.d. samples. For example, estimation of limiting variances and covariances is not needed. A Monte Carlo simulation study is presented to compare the new method with the normal approximation method and an existing EL 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.     

Partial covariate adjusted regression

Covariate adjusted regression (CAR) is a recently proposed adjustment method for regression analysis where both the response and predictors are not directly observed [?entürk, D., Müller, H.G., 2005. Covariate adjusted regression. Biometrika 92, 75–89]. The available data have been distorted by unknown functions of an observable confounding covariate. CAR provides consistent estimators for the coefficients of the regression between the variables of interest, adjusted for the confounder. We develop a broader class of partial covariate adjusted regression (PCAR) models to accommodate both distorted and undistorted (adjusted/unadjusted) predictors. The PCAR model allows for unadjusted predictors, such as age, gender and demographic variables, which are common in the analysis of biomedical and epidemiological data. The available estimation and inference procedures for CAR are shown to be invalid for the proposed PCAR model. We propose new estimators and develop new inference tools for the more general PCAR setting. In particular, we establish the asymptotic normality of the proposed estimators and propose consistent estimators of their asymptotic variances. Finite sample properties of the proposed estimators are investigated using simulation studies and the method is also illustrated with a Pima Indians diabetes data set.     

Confidence interval of the jump activity index based on empirical likelihood using high frequency data

It is widely accepted that jumps exist in the asset price process. The jump activity index is a natural measure of how frequent the jumps are. Statistical inference of the jump activity index is of importance in determining the type of process that underlies the dynamics of the log price process. In this paper, we implement the empirical likelihood approach to construct the confidence interval of the jump activity index of a pure jump model using high frequency data. Wilks' theorem is established. We also extend the result on Zhao and Wu (2009)'s estimator to the more general framework in this paper. Simulation studies demonstrate the good performance of the empirical likelihood approach. Compared with the existing non-parametric estimator proposed by Zhao and Wu (2009), the empirical likelihood approach gives more accurate coverage probabilities in the simulation studies.     

Nonparametric estimation of linear functionals of a bivariate distribution under univariate censoring

In the presence of univariate censoring, a class of nonparametric estimators is proposed for linear functionals of a bivariate distribution of paired failure times. The estimators are shown to be root-n consistent and asymptotically normal. An adjusted empirical log-likelihood ratio statistic is developed and proved to follow a chi-square distribution asymptotically. Two types of confidence intervals, based on the normal approximation method and the empirical likelihood method, respectively, are constructed to make inference about the linear functionals. Their performance is evaluated in several simulation studies and a real example.     

Semi-parametric hybrid empirical likelihood inference for two-sample comparison with censored data

Two-sample comparison problems are often encountered in practical projects and have widely been studied in literature. Owing to practical demands, the research for this topic under special settings such as a semiparametric framework have also attracted great attentions. Zhou and Liang (Biometrika 92:271–282, 2005) proposed an empirical likelihood-based semi-parametric inference for the comparison of treatment effects in a two-sample problem with censored data. However, their approach is actually a pseudo-empirical likelihood and the method may not be fully efficient. In this study, we develop a new empirical likelihood-based inference under more general framework by using the hazard formulation of censored data for two sample semi-parametric hybrid models. We demonstrate that our empirical likelihood statistic converges to a standard chi-squared distribution under the null hypothesis. We further illustrate the use of the proposed test by testing the ROC curve with censored data, among others. Numerical performance of the proposed method is also examined.