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.     

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.     

Inference for the Mean Difference in the Two-Sample Random Censorship Model

Inference for the mean difference in the two-sample random censorship model is an important problem in comparative survival and reliability test studies. This paper develops an adjusted empirical likelihood inference and a martingale-based bootstrap inference for the mean difference. A nonparametric version of Wilks' theorem for the adjusted empirical likelihood is derived, and the corresponding empirical likelihood confidence interval of the mean difference is constructed. Also, it is shown that the martingale-based bootstrap gives a correct first order asymptotic approximation of the corresponding estimator of the mean difference, which ensures that the martingale-based bootstrap confidence interval has asymptotically correct coverage probability. A simulation study is conducted to compare the adjusted empirical likelihood, the martingale-based bootstrap, and Efron's bootstrap in terms of coverage accuracies and average lengths of the confidence intervals. The simulation indicates that the proposed adjusted empirical likelihood and the martingale-based bootstrap confidence procedures are comparable, and both seem to outperform Efron's bootstrap procedure.     

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 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.     

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     

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.     

Empirical likelihood-based subset selection for partially linear autoregressive models

Based on the empirical likelihood method, the subset selection and hypothesis test for parameters in a partially linear autoregressive model are investigated. We show that the empirical log-likelihood ratio at the true parameters converges to the standard chi-square distribution. We then present the definitions of the empirical likelihood-based Bayes information criteria （EBIC） and Akaike information criteria （EAIC）. The results show that EBIC is consistent at selecting subset variables while EAIC is not. Simulation studies demonstrate that the proposed empirical likelihood confidence regions have better coverage probabilities than the least square method, while EBIC has a higher chance to select the true model than EAIC.     

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.     

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.     

Likelihood Based Confidence Intervals for the Tail Index

For the estimation of the tail index of a heavy tailed distribution, one of the well-known estimators is the Hill estimator (Hill, 1975). One obvious way to construct a confidence interval for the tail index is via the normal approximation of the Hill estimator. In this paper we apply both the empirical likelihood method and the parametric likelihood method to obtaining confidence intervals for the tail index. Our limited simulation study indicates that the normal approximation method is worse than the other two methods in terms of coverage probability, and the empirical likelihood method and the parametric likelihood method are comparable.     

Bootstrap and empirical likelihood methods in extremes

One of the major interests in extreme-value statistics is to infer the tail properties of the distribution functions in the domain of attraction of an extreme-value distribution and to predict rare events. In recent years, much effort in developing new methodologies has been made by many researchers in this area so as to diminish the impact of the bias in the estimation and achieve some asymptotic optimality in inference problems such as estimating the optimal sample fractions and constructing confidence intervals of various quantities. In particular, bootstrap and empirical likelihood methods, which have been widely used in many areas of statistics, have drawn attention. This paper reviews some novel applications of the bootstrap and the empirical likelihood techniques in extreme-value statistics. Dedicated to Professor Laurens de Haan on the occasion of his 70th birthday.     

Maximum Kernel Likelihood Estimation

We introduce an estimator for the population mean based on maximizing likelihoods formed by parameterizing a kernel density estimate. Due to these origins, we have dubbed the estimator the maximum kernel likelihood estimate (MKLE). A speedy computational method to compute the MKLE based on binning is implemented in a simulation study which shows that the MKLE at an optimal bandwidth is decidedly superior in terms of efficiency to the sample mean and other measures of location for heavy tailed symmetric distributions. An empirical rule and a computational method to estimate this optimal bandwidth are developed and used to construct bootstrap confidence intervals for the population mean. We show that the intervals have approximately nominal coverage and have significantly smaller average width than the standard t and z intervals. Finally, we develop some mathematical properties for a very close approximation to the MKLE called the kernel mean. In particular, we demonstrate that the kernel mean is indeed unbiased for the population mean for symmetric distributions.     

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.     

具有随机右删失随机变量分位数的经验似然推断

本文研究了具有随机右删失随机变量分位数的置信域的构造.利用经验似然和截尾值估算相结合的方法,给出了分位数的对数经验似然比统计量,在较少的条件下证明了该统计量的极限分布为自由度为1的x~2分布.使得完全数据下的分位数的经验似然推断方法应用到非完全数据中.     

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.     

Penalized empirical likelihood estimation of semiparametric models

We propose an empirical likelihood-based estimation method for conditional estimating equations containing unknown functions, which can be applied for various semiparametric models. The proposed method is based on the methods of conditional empirical likelihood and penalization. Thus, our estimator is called the penalized empirical likelihood (PEL) estimator. For the whole parameter including infinite-dimensional unknown functions, we derive the consistency and a convergence rate of the PEL estimator. Furthermore, for the finite-dimensional parametric component, we show the asymptotic normality and efficiency of the PEL estimator. We illustrate the theory by three examples. Simulation results show reasonable finite sample properties of our estimator.     

Empirical likelihood inference for the accelerated failure time model

Accelerated failure time (AFT) models are useful regression tools for studying the association between a survival time and covariates. Semiparametric inference procedures have been proposed in an extensive literature. Among these, use of an estimating equation which is monotone in the regression parameter and has some excellent properties was proposed by Fygenson and Ritov (1994). However, there is a serious under-coverage problem for small sample sizes. In this paper, we derive the limiting distribution of the empirical log-likelihood ratio for the regression parameter on the basis of the monotone estimating equations. Furthermore, the empirical likelihood (EL) confidence intervals/regions for the regression parameter are obtained. We conduct a simulation study in order to compare the proposed EL method with the normal approximation method. The simulation results suggest that the empirical likelihood based method outperforms the normal approximation based method in terms of coverage probability. Thus, the proposed EL method overcomes the under-coverage problem of the normal approximation method.     

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.     

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.