Coverage Accuracy of Confidence Intervals in Nonparametric Regression

Point-wise confidence intervals for a nonparametric regression function with random design points are considered. The confidence intervals are those based on the traditional normal approximation and the empirical likelihood. Their coverage accuracy is assessed by developing the Edgeworth expansions for the coverage probabilities. It is shown that the empirical likelihood confidence intervals are Bartlett correctable.     

条件密度置信区间的覆盖精度

本文用经验似然方法讨论了条件密度的置信区间的构造. 通过对覆盖概率的Edgeworth展开得到了经验似然置信区间的覆盖精度, 同时证明了条件密度的经验似然置信区间的Bartlett可修正性     

Empirical likelihood for linear models with missing responses

The purpose of this article is to use an empirical likelihood method to study the construction of confidence intervals and regions for the parameters of interest in linear regression models with missing response data. A class of empirical likelihood ratios for the parameters of interest are defined such that any of our class of ratios is asymptotically chi-squared. Our approach is to directly calibrate the empirical log-likelihood ratio, and does not need multiplication by an adjustment factor for the original ratio. Also, a class of estimators for the parameters of interest is constructed, and the asymptotic distributions of the proposed estimators are obtained. Our results can be used directly to construct confidence intervals and regions for the parameters of interest. 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 confidence intervals for hazard and density functions under right censorship

In this paper, we use smoothed empirical likelihood methods to construct confidence intervals for hazard and density functions under right censorship. Some empirical log-likelihood ratios for the hazard and density functions are obtained and their asymptotic limits are derived. Approximate confidence intervals based on these methods are constructed. Simulation studies are used to compare the empirical likelihood methods and the normal approximation methods in terms of coverage accuracy. It is found that the empirical likelihood methods provide better inference.     

Empirical likelihood-based inferences for the Lorenz curve

In this paper, we discuss empirical likelihood-based inferences for the Lorenz curve. The profile empirical likelihood ratio statistics for the Lorenz ordinate are defined under the simple random sampling and the stratified random sampling designs. It is shown that the limiting distributions of the profile empirical likelihood ratio statistics are scaled Chi-square distributions with one degree of freedom. We also derive the limiting processes of the associated empirical likelihood-based Lorenz processes. Hybrid bootstrap and empirical likelihood intervals for the Lorenz ordinate are proposed based on the newly developed empirical likelihood theory. Extensive simulation studies are conducted to compare the relative performances of various confidence intervals for Lorenz ordinates in terms of coverage probability and average interval length. The finite sample performances of the empirical likelihood-based confidence bands are also illustrated in simulation studies. Finally, a real example is used to illustrate the application of the recommended intervals.     

Empirical likelihood method for quantiles with response data missing at random

Empirical likelihood is a nonparametric method for constructing confidence intervals and tests,notably in enabling the shape of a confidence region determined by the sample data.This paper presents a new version of the empirical likelihood method for quantiles under kernel regression imputation to adapt missing response data.It eliminates the need to solve nonlinear equations,and it is easy to apply.We also consider exponential empirical likelihood as an alternative method.Numerical results are presented to compare our method with others.     

EMPIRICAL LIKELIHOOD RATIO CONFIDENCE INTERVALS FOR VARIOUS DIFFERENCES OF TWO POPULATIONS

Recently the empirical likelihood has been shown to be very useful in nonparametric models. Qin combined the empirical likelihood thought and the parametric likelihood method to construct confidence intervals for the difference of two population means in a semiparametric model. In this paper, we use the empirical likelihood thought to construct confidence intervals for some differences of two populations in a nonparametric model. A version of Wilks' theorem is developed.     

Empirical likelihood for the contrast of two hazard functions with right censoring

In this paper, we consider the standard two-sample framework with right censoring. We construct useful confidence intervals for the ratio or difference of two hazard functions using smoothed empirical likelihood (EL) methods. The empirical log-likelihood ratio is derived and its asymptotic distribution is a standard chi-squared distribution. Bootstrap confidence bands are also proposed. Simulation studies show that the proposed EL confidence intervals have outperformed normal approximation methods in terms of coverage probability. It is concluded that the empirical likelihood methods provide better inference results.     

Empirical likelihood inference for diffusion processes with jumps

In this paper, we consider the empirical likelihood inference for the jump-diffusion model. We construct the confidence intervals based on the empirical likelihood for the infinitesimal moments in the jump-diffusion models. They are better than the confidence intervals which are based on the asymptotic normality of point estimates.     

On the tail index of a heavy tailed distribution

This paper proposes some new estimators for the tail index of a heavy tailed distribution when only a few largest values are observed within blocks. These estimators are proved to be asymptotically normal under suitable conditions, and their Edgeworth expansions are obtained. Empirical likelihood method is also employed to construct confidence intervals for the tail index. The comparison for the confidence intervals based on the normal approximation and the empirical likelihood method is made in terms of coverage probability and length of the confidence intervals. The simulation study shows that the empirical likelihood method outperforms the normal approximation method.     

Joint empirical likelihood confidence regions for a finite number of quantiles under strong mixing samples

In this paper, we obtain the joint empirical likelihood confidence regions for a finite number of quantiles under strong mixing samples. As an application of this result, the empirical likelihood confidence intervals for the difference of any two quantiles are also obtained.     

广义Lorenz 曲线的非参数统计推断

本文讨论了广义Lorenz 曲线的经验似然统计推断. 在简单随机抽样、分层随机抽样和整群随机抽样下, 本文分别定义了广义Lorenz 坐标的pro le 经验似然比统计量, 得出这些经验似然比的极限分布为带系数的自由度为1 的χ² 分布. 对于整个Lorenz 曲线, 基于经验似然方法类似地得出相应的极限过程. 根据所得的经验似然理论, 本文给出了bootstrap 经验似然置信区间构造方法, 并通过数据模拟, 对新给出的广义Lorenz 坐标的bootstrap 经验似然置信区间与渐近正态置信区间以及bootstrap 置信区间等进行了对比研究. 对整个Lorenz 曲线, 基于经验似然方法对其置信域也进行了模拟研究. 最后我们将所推荐的置信区间应用到实例中.     

Confidence Intervals and Accuracy Estimation for Heavy-Tailed Generalized Pareto Distributions

The generalized Pareto distribution (GPD) is a two-parameter family of distributions which can be used to model exceedances over a threshold. We compare the empirical coverage of some standard bootstrap and likelihood-based confidence intervals for the parameters and upper p-quantiles of the GPD. Simulation results indicate that none of the bootstrap methods give satisfactory intervals for small sample sizes. By applying a general method of D. N. Lawley, correction factors for likelihood ratio statistics of parameters and quantiles of the GPD have been calculated. Simulations show that for small sample sizes accuracy of confidence intervals can be improved by incorporating the computed correction factors to the likelihood-based confidence intervals. While the modified likelihood method has better empirical coverage probability, the mean length of produced intervals are not longer than corresponding bootstrap confidence intervals. This article also investigates the performance of some bootstrap methods for estimation of accuracy measures of maximum likelihood estimators of parameters and quantiles of the GPD.     

Empirical likelihood for semiparametric regression model with missing response data

A bias-corrected technique for constructing the empirical likelihood ratio is used to study a semiparametric regression model with missing response data. We are interested in inference for the regression coefficients, the baseline function and the response mean. A class of empirical likelihood ratio functions for the parameters of interest is defined so that undersmoothing for estimating the baseline function is avoided. The existing data-driven algorithm is also valid for selecting an optimal bandwidth. Our approach is to directly calibrate the empirical log-likelihood ratio so that the resulting ratio is asymptotically chi-squared. Also, a class of estimators for the parameters of interest is constructed, their asymptotic distributions are obtained, and consistent estimators of asymptotic bias and variance are provided. Our results can be used to construct confidence intervals and bands for the parameters of interest. A simulation study is undertaken to compare the empirical likelihood with the normal approximation-based method in terms of coverage accuracies and average lengths of confidence intervals. An example for an AIDS clinical trial data set is used for illustrating our methods.     

Model-based likelihood ratio confidence intervals for survival functions

We introduce an adjusted likelihood ratio procedure for computing pointwise confidence intervals for survival functions from censored data. The test statistic, scaled by a ratio of two variance quantities, is shown to converge to a chi-squared distribution with one degree of freedom. The confidence intervals are seen to be a neighborhood of a semiparametric survival function estimator and are shown to have correct empirical coverage. Numerical studies also indicate that the proposed intervals have smaller estimated mean lengths in comparison to the ones that are produced as a neighborhood of the Kaplan-Meier estimator. We illustrate our method using a lung cancer data set.     

相依序列密度函数的经验似然推断

本文利用了强平稳$m-$相依序列的特殊性质,讨论了$m-$相依序列密度函数的经验似然推断, 给出了似然比统计量的极限分布,可构造参数的经验似然置信区间. 并且通过模拟计算来说明有限样本下应用经验似然方法的合理性.     

Empirical likelihood for conditional quantile with left-truncated and dependent data

In this paper, we employ the method of empirical likelihood to construct confidence intervals for a conditional quantile in the presence and absence of auxiliary information, respectively, for the left-truncation model. It is proved that the empirical likelihood ratio admits a limiting chi-square distribution with one degree of freedom when the lifetime observations with multivariate covariates form a stationary α-mixing sequence. For the problem of testing a hypothesis on the conditional quantile, it is shown that the asymptotic power of the test statistic based on the empirical likelihood ratio with the auxiliary information is larger than that of the one based on the standard empirical likelihood ratio. The finite sample performance of the empirical likelihood confidence intervals in the presence and absence of auxiliary information is investigated through simulations.     

Empirical likelihood inference for a partially linear errors-in-variables model with covariate data missing at random

The authors study the empirical likelihood method for partially linear errors-in-variablesmodel with covariate data missing at random. Empirical likelihood ratios for the regression coefficients and the baseline function are investigated, and the corresponding empirical log-likelihood ratios are proved to be asymptotically standard chi-squared, which can be used to construct confidence regions. The finite sample behavior of the proposed methods is evaluated by a simulation study which indicates that the proposed methods are comparable in terms of coverage probabilities and average length of confidence intervals. Finally, the Earthquake Magnitude dataset is used to illustrate our proposed method.     

核实数据下响应变量缺失的线性EV模型经验似然推断

考虑响应变量随机缺失而协变量带有误差的线性模型,借助于核实数据和借补方法,构造了回归系数的两种经验似然比,证明了所提出的估计的经验对数似然比渐近于一个自由度为1的独立χ2变量的加权和;而经调整后所得的调整经验对数似然比渐近于自由度为p的χ2分布,该结果可以用来构造未知参数的置信域.此外,我们也构造了响应均值的调整经验对数似然比统计量,并证明了所提出的统计量渐近于x2分布,可用此结果构造响应均值的置信域.通过模拟研究比较了置信域的精度及其平均区间长度.     

Empirical likelihood for single-index models

The empirical likelihood method is especially useful for constructing confidence intervals or regions of the parameter of interest. This method has been extensively applied to linear regression and generalized linear regression models. In this paper, the empirical likelihood method for single-index regression models is studied. An estimated empirical log-likelihood approach to construct the confidence region of the regression parameter is developed. An adjusted empirical log-likelihood ratio is proved to be asymptotically standard chi-square. A simulation study indicates that compared with a normal approximation-based approach, the proposed method described herein works better in terms of coverage probabilities and areas (lengths) of confidence regions (intervals).