Empirical likelihood inference for a common mean in the presence of heteroscedasticity

The authors develop empirical likelihood (EL) based methods of inference for a common mean using data from several independent but nonhomogeneous populations. For point estimation, they propose a maximum empirical likelihood (MEL) estimator and show that it is n‐consistent and asymptotically optimal. For confidence intervals, they consider two EL based methods and show that both intervals have approximately correct coverage probabilities under large samples. Finite‐sample performances of the MEL estimator and the EL based confidence intervals are evaluated through a simulation study. The results indicate that overall the MEL estimator and the weighted EL confidence interval are superior alternatives to the existing methods.     

Pseudo‐empirical likelihood ratio confidence intervals for complex surveys

The authors show how an adjusted pseudo‐empirical likelihood ratio statistic that is asymptotically distributed as a chi‐square random variable can be used to construct confidence intervals for a finite population mean or a finite population distribution function from complex survey samples. They consider both non‐stratified and stratified sampling designs, with or without auxiliary information. They examine the behaviour of estimates of the mean and the distribution function at specific points using simulations calling on the Rao‐Sampford method of unequal probability sampling without replacement. They conclude that the pseudo‐empirical likelihood ratio confidence intervals are superior to those based on the normal approximation, whether in terms of coverage probability, tail error rates or average length of the intervals.     

Empirical Likelihood Inference for the Rao‐Hartley‐Cochran Sampling Design

The Hartley‐Rao‐Cochran sampling design is an unequal probability sampling design which can be used to select samples from finite populations. We propose to adjust the empirical likelihood approach for the Hartley‐Rao‐Cochran sampling design. The approach proposed intrinsically incorporates sampling weights, auxiliary information and allows for large sampling fractions. It can be used to construct confidence intervals. In a simulation study, we show that the coverage may be better for the empirical likelihood confidence interval than for standard confidence intervals based on variance estimates. The approach proposed is simple to implement and less computer intensive than bootstrap. The confidence interval proposed does not rely on re‐sampling, linearization, variance estimation, design‐effects or joint inclusion probabilities.     

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.     

Odds ratio inference from stratified samples

One of the common problems encountered in applied statistics is that of comparing two proportions from stratified samples. One approach to this problem is via inference on the corresponding odds ratio. In this paper, the various point and interval estimators of and hypothesis testing procedures for a common odds ratio from multiple 2 ×2 tables are reviewed. Based On research to date, the conditional maximum likelihood and Mantel-Haenszel estimators are recommended as the point estimators of choice. Neither confidence intervals nor hypothesis testing metthods have been studied as well as the point estimators, but there is a confidence interval method associated with the Mantel-Haenszel estimator that is a good choice.     

Weighted Likelihood for Semiparametric Models and Two-phase Stratified Samples, with Application to Cox Regression

Abstract.  We consider semiparametric models for which solution of Horvitz–Thompson or inverse probability weighted (IPW) likelihood equations with two-phase stratified samples leads to consistent and asymptotically Gaussian estimators of both Euclidean and non-parametric parameters. For Bernoulli (independent and identically distributed) sampling, standard theory shows that the Euclidean parameter estimator is asymptotically linear in the IPW influence function. By proving weak convergence of the IPW empirical process, and borrowing results on weighted bootstrap empirical processes, we derive a parallel asymptotic expansion for finite population stratified sampling. Several of our key results have been derived already for Cox regression with stratified case–cohort and more general survey designs. This paper is intended to help interpret this previous work and to pave the way towards a general Horvitz–Thompson approach to semiparametric inference with data from complex probability samples.     

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.     

Empirical Likelihood-based Inference in Linear Models with Missing Data

The missing response problem in linear regression is studied. An adjusted empirical likelihood approach to inference on the mean of the response variable is developed. A non-parametric version of Wilks's theorem for the adjusted empirical likelihood is proved, and the corresponding empirical likelihood confidence interval for the mean is constructed. With auxiliary information, an empirical likelihood-based estimator with asymptotic normality is defined and an adjusted empirical log-likelihood function with asymptotic χ² is derived. A simulation study is conducted to compare the adjusted empirical likelihood methods and the normal approximation methods in terms of coverage accuracies and average lengths of the confidence intervals. Based on biases and standard errors, a comparison is also made between the empirical likelihood-based estimator and related estimators by simulation. Our simulation indicates that the adjusted empirical likelihood methods perform competitively and the use of auxiliary information provides improved inferences.     

Censored median regression and profile empirical likelihood

We implement profile empirical likelihood-based inference for censored median regression models. Inference for any specified subvector is carried out by profiling out the nuisance parameters from the “plug-in” empirical likelihood ratio function proposed by Qin and Tsao. To obtain the critical value of the profile empirical likelihood ratio statistic, we first investigate its asymptotic distribution. The limiting distribution is a sum of weighted chi square distributions. Unlike for the full empirical likelihood, however, the derived asymptotic distribution has intractable covariance structure. Therefore, we employ the bootstrap to obtain the critical value, and compare the resulting confidence intervals with the ones obtained through Basawa and Koul’s minimum dispersion statistic. Furthermore, we obtain confidence intervals for the age and treatment effects in a lung cancer data set.     

Integrated Likelihood Inference in Small Sample Meta‐analysis for Continuous Outcomes

This paper proposes the use of the integrated likelihood for inference on the mean effect in small‐sample meta‐analysis for continuous outcomes. The method eliminates the nuisance parameters given by variance components through integration with respect to a suitable weight function, with no need to estimate them. The integrated likelihood approach takes into proper account the estimation uncertainty of within‐study variances, thus providing confidence intervals with empirical coverage closer to nominal levels than standard likelihood methods. The improvement is remarkable when either (i) the number of studies is small to moderate or (ii) the small sample size of the studies does not allow to consider the within‐study variances as known, as common in applications. Moreover, the use of the integrated likelihood avoids numerical pitfalls related to the estimation of variance components which can affect alternative likelihood approaches. The proposed methodology is illustrated via simulation and applied to a meta‐analysis study in nutritional science.     

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.     

Instrumental variable-based empirical likelihood inferences for varying-coefficient models with error-prone covariates

This paper presents the empirical likelihood inferences for a class of varying-coefficient models with error-prone covariates. We focus on the case that the covariance matrix of the measurement errors is unknown and neither repeated measurements nor validation data are available. We propose an instrumental variable-based empirical likelihood inference method and show that the proposed empirical log-likelihood ratio is asymptotically chi-squared. Then, the confidence intervals for the varying-coefficient functions are constructed. Some simulation studies and a real data application are used to assess the finite sample performance of the proposed empirical likelihood procedure.     

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 generalized partially linear varying-coefficient models

Generalized partially linear varying-coefficient models (GPLVCM) are frequently used in statistical modeling. However, the statistical inference of the GPLVCM, such as confidence region/interval construction, has not been very well developed. In this article, empirical likelihood-based inference for the parametric components in the GPLVCM is investigated. Based on the local linear estimators of the GPLVCM, an estimated empirical likelihood-based statistic is proposed. We show that the resulting statistic is asymptotically non-standard chi-squared. By the proposed empirical likelihood method, the confidence regions for the parametric components are constructed. In addition, when some components of the parameter are of particular interest, the construction of their confidence intervals is also considered. A simulation study is undertaken to compare the empirical likelihood and the other existing methods in terms of coverage accuracies and average lengths. The proposed method is applied to a real example.     

Empirical likelihood inference for the bivariate survival function under univariate censoring

In this article, we apply the empirical likelihood method to make inference on the bivariate survival function of paired failure times by estimating the survival function of censored time with the Kaplan–Meier estimator. Adjusted empirical likelihood (AEL) confidence intervals for the bivariate survival function are developed. We conduct a simulation study to compare the proposed AEL method with other methods. The simulation study shows the proposed AEL method has better performance than other existing methods. We illustrate the proposed method by analyzing the skin graft data.     

Empirical likelihood‐based inferences for a low income proportion

Low income proportion is an important index in comparisons of poverty in countries around the world. The stability of a society depends heavily on this index. An accurate and reliable estimation of this index plays an important role for government's economic policies. In this paper, the authors study empirical likelihood‐based inferences for a low income proportion under the simple random sampling and stratified random sampling designs. It is shown that the limiting distributions of the empirical likelihood ratios for the low income proportion are the scaled chi‐square distributions. The authors propose various empirical likelihood‐based confidence intervals for the low income proportion. Extensive simulation studies are conducted to evaluate the relative performance of the normal approximation‐based interval, bootstrap‐based intervals, and the empirical likelihood‐based intervals. The proposed methods are also applied to analyzing a real economic survey income dataset. The Canadian Journal of Statistics 39: 1–16; 2011 ©2011 Statistical Society of Canada     

Empirical likelihood-based inference for parameter and nonparametric function in partially nonlinear models

This paper is concerned with statistical inference for partially nonlinear models. Empirical likelihood method for parameter in nonlinear function and nonparametric function is investigated. The empirical log-likelihood ratios are shown to be asymptotically chi-square and then the corresponding confidence intervals are constructed. By the empirical likelihood ratio functions, we also obtain the maximum empirical likelihood estimators of the parameter in nonlinear function and nonparametric function, and prove the asymptotic normality. A simulation study indicates that, compared with normal approximation-based method and the bootstrap method, the empirical likelihood method performs better in terms of coverage probabilities and average length/widths of confidence intervals/bands. An application to a real dataset is illustrated.     

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.     

Confidence Intervals for the Median of a Finite Population Based on the Sign Test

This article presents methods for constructing confidence intervals for the median of a finite population under simple random sampling without replacement, stratified random sampling, and cluster sampling. The confidence intervals, as well as point estimates and test statistics, are derived from sign estimating functions which are based on the well-known sign test. Therefore, a unified approach for inference about the median of a finite population is given.     

关于捕获再捕获抽样的置信区间

捕获再捕获抽样是一种应用广泛的抽样方法。运用随机模拟,研究捕获再捕获抽样的三种估计量的均值、方差、偏度、峰度、利用近似正态分布构造的置信区间的统计性质。改进了估计量的样本方差计算公式,使得利用近似正态分布构造的置信区间更优。