Empirical Likelihood Inference for Longitudinal Data with Missing Response Variables and Error-Prone Covariates

We consider statistical inference for longitudinal partially linear models when the response variable is sometimes missing with missingness probability depending on the covariate that is measured with error. The block empirical likelihood procedure is used to estimate the regression coefficients and residual adjusted block empirical likelihood is employed for the baseline function. This leads us to prove a nonparametric version of Wilk's theorem. Compared with methods based on normal approximations, our proposed method does not require a consistent estimators for the asymptotic variance and bias. An application to a longitudinal study is used to illustrate the procedure developed here. A simulation study is also reported.     

The Block Empirical Likelihood Method of the Semivarying Coefficient Model with Application to Longitudinal Data

In this article, we consider a semivarying coefficient model with application to longitudinal data. In order to accommodate the within-group correlation, we apply the block empirical likelihood procedure to semivarying coefficient longitudinal data model, and prove a nonparametric version of Wilks' theorem which can be used to construct the block empirical likelihood confidence region with asymptotically correct coverage probability for the parametric component. In comparison with normal approximations, the proposed method does not require a consistent estimator for the asymptotic covariance matrix, making it easier to conduct inference for the model's parametric component. Simulations demonstrate how the proposed method works.     

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.     

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.     

Generalized Empirical Likelihood Inference in Generalized Linear Models for Longitudinal Data

In this article, the generalized linear model for longitudinal data is studied. A generalized empirical likelihood method is proposed by combining generalized estimating equations and quadratic inference functions based on the working correlation matrix. It is proved that the proposed generalized empirical likelihood ratios are asymptotically chi-squared under some suitable conditions, and hence it can be used to construct the confidence regions of the parameters. In addition, the maximum empirical likelihood estimates of parameters are obtained, and their asymptotic normalities are proved. Some simulations are undertaken to compare the generalized empirical likelihood and normal approximation-based method in terms of coverage accuracies and average areas/lengths of confidence regions/intervals. An example of a real data is used for illustrating our methods.     

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.     

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.     

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.     

Estimation and inference of combining quantile and least-square regressions with missing data

In this paper, we consider how to incorporate quantile information to improve estimator efficiency for regression model with missing covariates. We combine the quantile information with least-squares normal equations and construct an unbiased estimating equations (EEs). The lack of smoothness of the objective EEs is overcome by replacing them with smooth approximations. The maximum smoothed empirical likelihood (MSEL) estimators are established based on inverse probability weighted (IPW) smoothed EEs and their asymptotic properties are studied under some regular conditions. Moreover, we develop two novel testing procedures for the underlying model. The finite-sample performance of the proposed methodology is examined by simulation studies. A real example is used to illustrate our methods.     

Kernel-based global MLE of partial linear random effects models for longitudinal data

Random effects models have been playing a critical role for modelling longitudinal data. However, there are little studies on the kernel-based maximum likelihood method for semiparametric random effects models. In this paper, based on kernel and likelihood methods, we propose a pooled global maximum likelihood method for the partial linear random effects models. The pooled global maximum likelihood method employs the local approximations of the nonparametric function at a group of grid points simultaneously, instead of one point. Gaussian quadrature is used to approximate the integration of likelihood with respect to random effects. The asymptotic properties of the proposed estimators are rigorously studied. Simulation studies are conducted to demonstrate the performance of the proposed approach. We also apply the proposed method to analyse correlated medical costs in the Medical Expenditure Panel Survey data set.     

Empirical likelihood for truncation parameter in random truncation model

In this paper we use the empirical likelihood method to construct confidence interval for truncation parameter in random truncation model. The empirical log-likelihood ratio is derived and its asymptotic distribution is shown to be a weighted chi-square. Simulation studies are used to compare the confidence intervals based on empirical likelihood and those based on normal approximation. It is found that the empirical likelihood method provides improved confidence interval.     

Empirical likelihood inference for partial functional linear model with missing responses

In this paper, we consider the empirical likelihood inferences of the partial functional linear model with missing responses. Two empirical log-likelihood ratios of the parameters of interest are constructed, and the corresponding maximum empirical likelihood estimators of parameters are derived. Under some regularity conditions, we show that the proposed two empirical log-likelihood ratios are asymptotic standard Chi-squared. Thus, the asymptotic results can be used to construct the confidence intervals/regions for the parameters of interest. We also establish the asymptotic distribution theory of corresponding maximum empirical likelihood estimators. A simulation study indicates that the proposed methods are comparable in terms of coverage probabilities and average lengths of confidence intervals. An example of real data is also used to illustrate our proposed methods.     

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.     

A two-sample empirical likelihood ratio test based on samples entropy

Powerful entropy-based tests for normality, uniformity and exponentiality have been well addressed in the statistical literature. The density-based empirical likelihood approach improves the performance of these tests for goodness-of-fit, forming them into approximate likelihood ratios. This method is extended to develop two-sample empirical likelihood approximations to optimal parametric likelihood ratios, resulting in an efficient test based on samples entropy. The proposed and examined distribution-free two-sample test is shown to be very competitive with well-known nonparametric tests. For example, the new test has high and stable power detecting a nonconstant shift in the two-sample problem, when Wilcoxon’s test may break down completely. This is partly due to the inherent structure developed within Neyman-Pearson type lemmas. The outputs of an extensive Monte Carlo analysis and real data example support our theoretical results. The Monte Carlo simulation study indicates that the proposed test compares favorably with the standard procedures, for a wide range of null and alternative distributions.     

Estimation and prediction for spatial generalized linear mixed models using high order Laplace approximation

Estimation and prediction in generalized linear mixed models are often hampered by intractable high dimensional integrals. This paper provides a framework to solve this intractability, using asymptotic expansions when the number of random effects is large. To that end, we first derive a modified Laplace approximation when the number of random effects is increasing at a lower rate than the sample size. Second, we propose an approximate likelihood method based on the asymptotic expansion of the log-likelihood using the modified Laplace approximation which is maximized using a quasi-Newton algorithm. Finally, we define the second order plug-in predictive density based on a similar expansion to the plug-in predictive density and show that it is a normal density. Our simulations show that in comparison to other approximations, our method has better performance. Our methods are readily applied to non-Gaussian spatial data and as an example, the analysis of the rhizoctonia root rot data is presented.     

Empirical likelihood confidence regions in the single-index model with growing dimensions

This paper investigates statistical inference for the single-index model when the number of predictors grows with sample size. Empirical likelihood method for constructing confidence region for the index vector, which does not require a multivariate non parametric smoothing, is employed. However, the classical empirical likelihood ratio for this model does not remain valid because plug-in estimation of an infinite-dimensional nuisance parameter causes a non negligible bias and the diverging number of parameters/predictors makes the limit not chi-squared any more. To solve these problems, we define an empirical likelihood ratio based on newly proposed weighted estimating equations and show that it is asymptotically normal. Also we find that different weights used in the weighted residuals require, for asymptotic normality, different diverging rate of the number of predictors. However, the rate n^1/3, which is a possible fastest rate when there are no any other conditions assumed in the setting under study, is still attainable. A simulation study is carried out to assess the performance of our method.     

Empirical likelihood for the difference of quantiles under censorship

In this paper, we use a smoothed empirical likelihood method to investigate the difference of quantiles under censorship. An empirical log-likelihood ratio is derived and its asymptotic distribution is shown to be chi-squared. Approximate confidence regions based on this method are constructed. Simulation studies are used to compare the empirical likelihood and the normal approximation method in terms of its coverage accuracy. It is found that the empirical likelihood method provides a much better performance. The research is supported by NSFC (10231030) and RFDP.     

Empirical likelihood for the varying‐coefficient single‐index model

In this article the author investigates the application of the empirical‐likelihood‐based inference for the parameters of varying‐coefficient single‐index model (VCSIM). Unlike the usual cases, if there is no bias correction the asymptotic distribution of the empirical likelihood ratio cannot achieve the standard chi‐squared distribution. To this end, a bias‐corrected empirical likelihood method is employed to construct the confidence regions (intervals) of regression parameters, which have two advantages, compared with those based on normal approximation, that is, (1) they do not impose prior constraints on the shape of the regions; (2) they do not require the construction of a pivotal quantity and the regions are range preserving and transformation respecting. A simulation study is undertaken to compare the empirical likelihood with the normal approximation in terms of coverage accuracies and average areas/lengths of confidence regions/intervals. A real data example is given to illustrate the proposed approach. The Canadian Journal of Statistics 38: 434–452; 2010 © 2010 Statistical Society of Canada     

Estimating the parameters of Poisson-exponential models

This paper proposes two methods of estimation for the parameters in a Poisson-exponential model. The proposed methods combine the method of moments with a regression method based on the empirical moment generating function. One of the methods is an adaptation of the mixed-moments procedure of Koutrouvelis & Canavos (1999). The asymptotic distribution of the estimator obtained with this method is derived. Finite-sample comparisons are made with the maximum likelihood estimator and the method of moments. The paper concludes with an exploratory-type analysis of real data based on the empirical moment generating function.     

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.