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.     

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 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.     

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 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 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.     

Approximate jackknife empirical likelihood method for estimating equations

It is known that the profile empirical likelihood method based on estimating equations is computationally intensive when the number of nuisance parameters is large. Recently, Li, Peng, & Qi (2011) proposed a jackknife empirical likelihood method for constructing confidence regions for the parameters of interest by estimating the nuisance parameters separately. However, when the estimators for the nuisance parameters have no explicit formula, the computation of the jackknife empirical likelihood method is still intensive. In this paper, an approximate jackknife empirical likelihood method is proposed to reduce the computation in the jackknife empirical likelihood method when the nuisance parameters cannot be estimated explicitly. A simulation study confirms the advantage of the new method. The Canadian Journal of Statistics 40: 110–123; 2012 © 2012 Statistical Society of Canada     

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 varying-coefficient semiparametric mixed-effects errors-in-variables models with longitudinal data

In this paper, the empirical likelihood inferences for varying-coefficient semiparametric mixed-effects errors-in-variables models with longitudinal data are investigated. We construct the empirical log-likelihood ratio function for the fixed-effects parameters and the mean parameters of random-effects. The empirical log-likelihood ratio at the true parameters is proven to be asymptotically $\chi ^2_{q+r}$ , where $q$ and $r$ are dimensions of the fixed and random effects respectively, and the corresponding confidence regions for them are then constructed. We also obtain the maximum empirical likelihood estimator of the parameters of interest, and prove it is the asymptotically normal under some suitable conditions. A simulation study and a real data application are undertaken to assess the finite sample performance of the proposed method.     

Empirical phi-divergence test statistics for testing simple and composite null hypotheses

The main purpose of this paper is to introduce first a new family of empirical test statistics for testing a simple null hypothesis when the vector of parameters of interest is defined through a specific set of unbiased estimating functions. This family of test statistics is based on a distance between two probability vectors, with the first probability vector obtained by maximizing the empirical likelihood (EL) on the vector of parameters, and the second vector defined from the fixed vector of parameters under the simple null hypothesis. The distance considered for this purpose is the phi-divergence measure. The asymptotic distribution is then derived for this family of test statistics. The proposed methodology is illustrated through the well-known data of Newcomb's measurements on the passage time for light. A simulation study is carried out to compare its performance with that of the EL ratio test when confidence intervals are constructed based on the respective statistics for small sample sizes. The results suggest that the ‘empirical modified likelihood ratio test statistic’ provides a competitive alternative to the EL ratio test statistic, and is also more robust than the EL ratio test statistic in the presence of contamination in the data. Finally, we propose empirical phi-divergence test statistics for testing a composite null hypothesis and present some asymptotic as well as simulation results for evaluating the performance of these test procedures.     

Empirical Likelihood for a Heteroscedastic Partial Linear Errors-in-Variables Model

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 heteroscedastic partially linear errors-in-variables model with martingale difference errors. 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.     

Inference from PseudoLikelihoods with Plug‐In Estimates

Effective implementation of likelihood inference in models for high‐dimensional data often requires a simplified treatment of nuisance parameters, with these having to be replaced by handy estimates. In addition, the likelihood function may have been simplified by means of a partial specification of the model, as is the case when composite likelihood is used. In such circumstances tests and confidence regions for the parameter of interest may be constructed using Wald type and score type statistics, defined so as to account for nuisance parameter estimation or partial specification of the likelihood. In this paper a general analytical expression for the required asymptotic covariance matrices is derived, and suggestions for obtaining Monte Carlo approximations are presented. The same matrices are involved in a rescaling adjustment of the log likelihood ratio type statistic that we propose. This adjustment restores the usual chi‐squared asymptotic distribution, which is generally invalid after the simplifications considered. The practical implication is that, for a wide variety of likelihoods and nuisance parameter estimates, confidence regions for the parameters of interest are readily computable from the rescaled log likelihood ratio type statistic as well as from the Wald type and score type statistics. Two examples, a measurement error model with full likelihood and a spatial correlation model with pairwise likelihood, illustrate and compare the procedures. Wald type and score type statistics may give rise to confidence regions with unsatisfactory shape in small and moderate samples. In addition to having satisfactory shape, regions based on the rescaled log likelihood ratio type statistic show empirical coverage in reasonable agreement with nominal confidence levels.     

Empirical likelihood ratio test for or against a set of inequality constraints

We use the empirical likelihood ratio approach introduced by Owen (Biometrika 75 (1988), 237–249) to test for or against a set of inequality constraints when the parameters are defined by estimating functions. Our objective in this paper is to show that under fairly general conditions, the limiting distributions of the empirical likelihood ratio test statistics are of chi-bar square type (as in the parametric case) and give the expression of the weighting values. The results obtained here are similar to those in El Barmi and Dykstra (1995) where a full distributional model is assumed. This work presents also an extension of the results in Qin and Lawless (1995).     

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.     

A note on empirical likelihoods derived from pairwise score functions

Pairwise likelihood functions are convenient surrogates for the ordinary likelihood, useful when the latter is too difficult or even impractical to compute. One drawback of pairwise likelihood inference is that, for a multidimensional parameter of interest, the pairwise likelihood analogue of the likelihood ratio statistic does not have the standard chi-square asymptotic distribution. Invoking the theory of unbiased estimating functions, this paper proposes and discusses a computationally and theoretically attractive approach based on the derivation of empirical likelihood functions from the pairwise scores. This approach produces alternatives to the pairwise likelihood ratio statistic, which allow reference to the usual asymptotic chi-square distribution and which are useful when the elements of the Godambe information are troublesome to evaluate or in the presence of large data sets with relative small sample sizes. Two Monte Carlo studies are performed in order to assess the finite-sample performance of the proposed empirical pairwise likelihoods.     

Empirical likelihood inference for the regression model of mean quality‐adjusted lifetime with censored data

The authors consider the empirical likelihood method for the regression model of mean quality‐adjusted lifetime with right censoring. They show that an empirical log‐likelihood ratio for the vector of the regression parameters is asymptotically a weighted sum of independent chi‐squared random variables. They adjust this empirical log‐likelihood ratio so that the limiting distribution is a standard chi‐square and construct corresponding confidence regions. Simulation studies lead them to conclude that empirical likelihood methods outperform the normal approximation methods in terms of coverage probability. They illustrate their methods with a data example from a breast cancer clinical trial study.     

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.     

Corrected likelihood-ratio tests in logistic regression using small-sample data

Likelihood-ratio tests (LRTs) are often used for inferences on one or more logistic regression coefficients. Conventionally, for given parameters of interest, the nuisance parameters of the likelihood function are replaced by their maximum likelihood estimates. The new function created is called the profile likelihood function, and is used for inference from LRT. In small samples, LRT based on the profile likelihood does not follow χ² distribution. Several corrections have been proposed to improve LRT when used with small-sample data. Additionally, complete or quasi-complete separation is a common geometric feature for small-sample binary data. In this article, for small-sample binary data, we have derived explicitly the correction factors of LRT for models with and without separation, and proposed an algorithm to construct confidence intervals. We have investigated the performances of different LRT corrections, and the corresponding confidence intervals through simulations. Based on the simulation results, we propose an empirical rule of thumb on the use of these methods. Our simulation findings are also supported by real-world data.     

Empirical Likelihood for an Autoregressive Model with Explanatory Variables

In this article, we use the empirical likelihood method to construct the confidence region for parameters in autoregressive model with martingale difference error. It is shown that the empirical log-likelihood ratio at the true parameter converges to the standard chi-square distribution. The simulation results suggest that the empirical likelihood method outperforms the normal approximation based method in terms of coverage probability.     

Empirical likelihood for single-index varying-coefficient models with right-censored data

This paper is concerned with an estimation procedure of a class of single-index varying-coefficient models with right-censored data. An adjusted empirical log-likelihood ratio for the index parameters, which are of primary interest, is proposed using a synthetic data approach. The adjusted empirical likelihood is shown to have a standard chi-squared limiting distribution. Furthermore, we increase the accuracy of the proposed confidence regions by using the constraint that the index is of norm 1. Simulation studies are carried out to highlight the performance of the proposed method compared with the traditional normal approximation method.