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.     

Empirical likelihood for AR-ARCH models based on LAD estimation

By employing the empirical likelihood method,confidence regions for the stationary AR(p)-ARCH(q) models are constructed.A self-weighted LAD estimator is proposed under weak moment conditions.An empirical log-likelihood ratio statistic is derived and its asymptotic distribution is obtained.Simulation studies show that the performance of empirical likelihood method is better than that of normal approximation of the LAD estimator in terms of the coverage accuracy,especially for relative small size of observation.     

Empirical Likelihood Inference for Functional Coefficient ARCH-M Model

Empirical likelihood inference for parametric and nonparametric parts in functional coefficient ARCH-M models is investigated in this paper. Firstly, the kernel smoothing technique is used to estimate coefficient function δ(x). In this way we obtain an estimated function with parameter β.Secondly, the empirical likelihood method is developed to estimate the parameter β. An estimated empirical log-likelohood ratio is proved to be asymptotically standard chi-squred, and the maximum empirical likelihood estimation(MELE) for β is shown to be asymptotically normal. Finally, based on the MELE of β, the empirical likelihood approach is again applied to reestimate the nonparametric part δ(x). The empirical log-likelohood ratio for δ(x) is proved to be also asymptotically standard chi-squred. Simulation study shows that the proposed method works better than the normal approximation method in terms of average areas of confidence regions for β, and the empirical likelihood confidence belt for δ(x) performs well.     

Empirical Likelihood Inference for AR（p） Model

In this article we study the empirical likelihood inference for AR（p） model. We propose the moment restrictions, by which we get the empirical likelihood estimator of the model parametric, and we also propose an empirical log-likelihood ratio base on this estimator. Our result shows that the EL estimator is asymptotically normal, and the empirical log-likelihood ratio is proved to be asymptotically standard chi-squared.     

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 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.     

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 for the smoothed LAD estimator in infinite variance autoregressive models

This paper proposes an empirical likelihood method to estimate the parameters of infinite variance autoregressive (IVAR) models and to construct confidence regions for the parameters. Simulation studies suggest that in small sample case, the empirical likelihood confidence regions may be more accurate than the confidence regions constructed by the normal approximation based on the self-weighted LAD estimator proposed by Ling (2005).     

Empirical likelihood for LAD estimators in infinite variance ARMA models

In this paper, we use an empirical likelihood method to construct confidence regions for the stationary ARMA(p,q) models with infinite variance. An empirical log-likelihood ratio is derived by the estimating equation of the self-weighted LAD estimator. It is proved that the proposed statistic has an asymptotic standard chi-squared distribution. Simulation studies show that in a small sample case, the performance of empirical likelihood method is better than that of normal approximation of the LAD estimator in terms of the coverage accuracy.     

沪深股市大盘指数收益率分布尾部的实证研究

通过H ill估计的改进方法对上证综合指数和深圳成分指数的收益率分布的尾部指数进行了参数估计,用χ2检验验证了指数的稳定性及其置信区间.在此基础上提出用尾部指数估计尾概率,达到风险控制的目的.实证研究表明,沪深大盘指数收益率分布具有肥尾的特征,但并不服从无限方差分布.     

Empirical likelihood for heteroscedastic partially linear models

We make empirical-likelihood-based inference for the parameters in heteroscedastic partially linear models. Unlike the existing empirical likelihood procedures for heteroscedastic partially linear models, the proposed empirical likelihood is constructed using components of a semiparametric efficient score. We show that it retains the double robustness feature of the semiparametric efficient estimator for the parameters and shares the desirable properties of the empirical likelihood for linear models. Compared with the normal approximation method and the existing empirical likelihood methods, the empirical likelihood method based on the semiparametric efficient score is more attractive not only theoretically but empirically. Simulation studies demonstrate that the proposed empirical likelihood provides smaller confidence regions than that based on semiparametric inefficient estimating equations subject to the same coverage probabilities. Hence, the proposed empirical likelihood is preferred to the normal approximation method as well as the empirical likelihood method based on semiparametric inefficient estimating equations, and it should be useful in practice.     

Empirical likelihood for median regression model with designed censoring variables

We propose a new and simple estimating equation for the parameters in median regression models with designed censoring variables, and then apply the empirical log likelihood ratio statistic to construct confidence region for the parameters. The empirical log likelihood ratio statistic is shown to have a standard chi-square distribution, which makes this method easy to implement. At the same time, another empirical log likelihood ratio statistic is proposed based on an existing estimating equation and the limiting distribution of the empirical likelihood ratio statistic is shown to be a sum of weighted chi-square distributions. We compare the performance of the empirical likelihood confidence region based on the new estimating equation, with that based on the existing estimating equation and a normal approximation method by simulation studies.     

Empirical Likelihood Inference Under Stratified Random Sampling in the Presence of Measurement Error

Suppose that several different imperfect instruments and one perfect instrument are used independently to measure some characteristic of a population. In order to make full use of the sample information, in this paper the empirical likelihood method is put forward for making inferences on parameters of interest under stratified random sampling in the presence of measurement error, Our results show that it can lead to estimators which are asymptotically normal and utilize all the available sample information. We also obtain the asymptotic distribution of empirical likelihood testing statistics. In particular, we apply the method to obtain estimator and confidence interval of population mean.     

On large deviation expansion of distribution of maximum likelihood estimator and its application in large sample estimation

For estimating an unknown parameter , the likelihood principle yields the maximum likelihood estimator. It is often favoured especially by the applied statistician, for its good properties in the large sample case. In this paper, a large deviation expansion for the distribution of the maximum likelihood estimator is obtained. The asymptotic expansion provides a useful tool to approximate the tail probability of the maximum likelihood estimator and to make statistical inference. Theoretical and numerical examples are given. Numerical results show that the large deviation approximation performs much better than the classical normal approximation.This work is supported in part by the Natural Science and Engineering Research Council of Canada under grant NSERC A-9216.This author is also partially supported by the National Science Foundation of China.     

Empirical likelihood based inference for semiparametric varying coefficient partially linear models with error-prone linear covariates

This paper considers statistical inference for semiparametric varying coefficient partially linear models with error-prone linear covariates. An empirical likelihood based statistic for parametric component is developed to construct confidence regions. The resulting statistic is shown to be asymptotically chi-square distributed. By the empirical likelihood ratio function, the maximum empirical likelihood estimator of the parameter is defined and the asymptotic normality is shown. A simulation experiment is conducted to compare the empirical likelihood, normal based and the naive empirical likelihood methods in terms of coverage accuracies of confidence regions.     

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 for the parametric part in partially linear errors-in-function models

Partially linear errors-in-function models were proposed by Liang (2000), but their inferences have not been systematically studied. This article proposes an empirical likelihood method to construct confidence regions of the parametric components. Under mild regularity conditions, the nonparametric version of the Wilk’s theorem is derived. Simulation studies show that the proposed empirical likelihood method provides narrower confidence regions, as well as higher coverage probabilities than those based on the traditional normal approximation method.     

区间数据均值的经验似然估计

本文提出了估计区间数据均值的经验似然方法,通过构造区间数据的无偏转换,导出了渐近服从χ2分布的对数经验似然函数,从而得到了均值的置信区间.通过若干模拟例子说明,用本文提出的方法得到的估计,优于用渐近正态法得到的估计.     

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 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.