Empirical Log-likelihood Ratio Statistic Research Articles

Empirical Likelihood for a Partially Linear Single-Index Measurement Error Model with Right-Censored Data

This article considers the application of the empirical likelihood method to a partially linear single-index measure error model (PLSIMeM) with right-censored data. With a synthetic data approach, an empirical log-likelihood ratio statistic for the parametric components is defined and it is shown that its limiting distribution is a mixture of central chi-squared distributions. Based on this, a confidence region for the parameters is constructed. To increase the accuracy of the confidence region, we also propose an adjusted empirical log-likelihood ratio statistic for the parameters. The resulting statistic is proved to follow a standard chi-squared limiting distribution. Simulation studies are undertaken to assess the finite sample performance of the proposed confidence regions.

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 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 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 for semi-parametric varying-coefficient partially linear EV models

In this paper, we apply empirical likelihood method to study the semi-parametric varying-coefficient partially linear errors-in-variables models. 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 under some suitable conditions, and hence it can be used to construct the confidence region for the parameter β. Some simulations indicate that, in terms of coverage probabilities and average lengths of the confidence intervals, the proposed method performs better than the 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 suitable conditions.

Saddlepoint Approximation for Sample Quantiles with Some Applications

In this article, we use the integral form of the binomial distribution to derive saddlepoint approximations for sample quantiles. As an application, we present the calculation of the tail probability of the empirical log-likelihood ratio statistic for quantiles. Simulation results are also given to show that our approximations are extremely accurate.

Generalized empirical likelihood inference in semiparametric regression model for longitudinal data

In this paper, we consider the semiparametric regression model for longitudinal data. Due to the correlation within groups, a generalized empirical log-likelihood ratio statistic for the unknown parameters in the model is suggested by introducing the working covariance matrix. It is proved that the proposed statistic is asymptotically standard chi-squared under some suitable conditions, and hence it can be used to construct the confidence regions of the parameters. A simulation study is conducted to compare the proposed method with the generalized least squares method in terms of coverage accuracy and average lengths of the confidence intervals.

Empirical likelihood intervals for the population mean and quantiles based on balanced ranked set samples

We consider nonparametric interval estimation for the population mean and quantiles based on a ranked set sample. The asymptotic distributions of the empirical log likelihood ratio statistic for the mean and quantiles are derived. Interval estimates of the population mean and quantiles are obtained by inverting the likelihood ratio statistic. Simulations are carried out to investigate and compare the performance of the empirical likelihood intervals with other known intervals.

Empirical Likelihood Confidence Region for the Parameter in a Partially Linear Errors-in-Variables Model

In this article, a partially linear errors-in-variables model is considered, and empirical log-likelihood ratio statistic for the unknown parameter in the model is suggested. It is proved that the proposed statistic is asymptotically standard chi-square distribution under some suitable conditions, and hence it can be used to construct the confidence region of the parameter. A simulation study indicates that, in terms of coverage probabilities and average lengths of the confidence intervals, the proposed method performs better than the least-squares method.

Nonparametric estimation of linear functionals of a bivariate distribution under univariate censoring

In the presence of univariate censoring, a class of nonparametric estimators is proposed for linear functionals of a bivariate distribution of paired failure times. The estimators are shown to be root- n consistent and asymptotically normal. An adjusted empirical log-likelihood ratio statistic is developed and proved to follow a chi-square distribution asymptotically. Two types of confidence intervals, based on the normal approximation method and the empirical likelihood method, respectively, are constructed to make inference about the linear functionals. Their performance is evaluated in several simulation studies and a real example.

Testing serial correlation in semiparametric varying-coefficient partially linear EV models

This paper studies estimation and serial correlation test of a semiparametric varying-coefficient partially linear EV model of the form Y = Xτβ + Zτα(T) + e, ξ = X + η with the identifying condition E[(e, ητ)τ] = 0, Cov[(e, ητ)τ] = σ2Ip+1. The estimators of interested regression parameters β, and the model error variance σ2, as well as the nonparametric components α(T), are constructed. Under some regular conditions, we show that the estimators of the unknown vector β and the unknown parameter σ2 are strongly consistent and asymptotically normal and that the estimator of α(T) achieves the optimal strong convergence rate of the usual nonparametric regression. Based on these estimators and asymptotic properties, we propose the VN,p test statistic and empirical log-likelihood ratio statistic for testing serial correlation in the model. The proposed statistics are shown to have asymptotic normal or chi-square distributions under the null hypothesis of no serial correlation. Some simulation studies are conducted to illustrate the finite sample performance of the proposed tests.

Empirical likelihood confidence regions of the parameters in a partially linear single-index model

In this paper, a partially linear single-index model is investigated, and three empirical log-likelihood ratio statistics for the unknown parameters in the model are suggested. It is proved that the proposed statistics are asymptotically standard chi-square under some suitable conditions, and hence can be used to construct the confidence regions of the parameters. Our methods can also deal with the confidence region construction for the index in the pure single-index model. A simulation study indicates that, in terms of coverage probabilities and average areas of the confidence regions, the proposed methods perform better than the least-squares method.

Empirical Likelihood for Partially Linear Models

In this paper, we consider the application of the empirical likelihood method to partially linear model. Unlike the usual cases, we first propose an approximation to the residual of the model to deal with the nonparametric part so that Owen's (1990) empirical likelihood approach can be applied. Then, under quite general conditions, we prove that the empirical log-likelihood ratio statistic is asymptotically chi-squared distributed. Therefore, the empirical likelihood confidence regions can be constructed accordingly.