Empirical likelihood for truncation parameter in random truncation model

In this paper we use the empirical likelihood method to construct confidence interval for truncation parameter in random truncation model. The empirical log-likelihood ratio is derived and its asymptotic distribution is shown to be a weighted chi-square. Simulation studies are used to compare the confidence intervals based on empirical likelihood and those based on normal approximation. It is found that the empirical likelihood method provides improved confidence interval.     

Jackknife Empirical Likelihood for the Accelerated Failure Time Model with Censored Data

Kendall and Gehan estimating functions are commonly used to estimate the regression parameter in accelerated failure time model with censored observations in survival analysis. In this paper, we apply the jackknife empirical likelihood method to overcome the computation difficulty about interval estimation. A Wilks’ theorem of jackknife empirical likelihood for U-statistic type estimating equations is established, which is used to construct the confidence intervals for the regression parameter. We carry out an extensive simulation study to compare the Wald-type procedure, the empirical likelihood method, and the jackknife empirical likelihood method. The proposed jackknife empirical likelihood method has a better performance than the existing methods. We also use a real data set to compare the proposed methods.     

Empirical likelihood for the difference of quantiles under censorship

In this paper, we use a smoothed empirical likelihood method to investigate the difference of quantiles under censorship. An empirical log-likelihood ratio is derived and its asymptotic distribution is shown to be chi-squared. Approximate confidence regions based on this method are constructed. Simulation studies are used to compare the empirical likelihood and the normal approximation method in terms of its coverage accuracy. It is found that the empirical likelihood method provides a much better performance. The research is supported by NSFC (10231030) and RFDP.     

Empirical Likelihood for Censored Linear Regression

In this paper we investigate the empirical likelihood method in a linear regression model when the observations are subject to random censoring. An empirical likelihood ratio for the slope parameter vector is defined and it is shown that its limiting distribution is a weighted sum of independent chi-square distributions. This reduces to the empirical likelihood to the linear regression model first studied by Owen (1991) if there is no censoring present. Some simulation studies are presented to compare the empirical likelihood method with the normal approximation based method proposed in Lai et al. (1995). It was found that the empirical likelihood method performs much better than the normal approximation method.     

Empirical Likelihood Based Synthetic Data Method for Censored Regression Analysis

In this article, we propose a new empirical likelihood method for linear regression analysis with a right censored response variable. The method is based on the synthetic data approach for censored linear regression analysis. A log-empirical likelihood ratio test statistic for the entire regression coefficients vector is developed and we show that it converges to a standard chi-squared distribution. The proposed method can also be used to make inferences about linear combinations of the regression coefficients. Moreover, the proposed empirical likelihood ratio provides a way to combine different normal equations derived from various synthetic response variables. Maximizing this empirical likelihood ratio yields a maximum empirical likelihood estimator which is asymptotically equivalent to the solution of the estimating equation that are optimal linear combination of the original normal equations. It improves the estimation efficiency. The method is illustrated by some Monte Carlo simulation studies as well as a real example.     

Empirical likelihood inference for parameters in a partially linear errors-in-variables model

In this paper, we consider the application of the empirical likelihood method to a partially linear model with measurement errors in the non-parametric part. It is shown that the empirical log-likelihood ratio at the true parameters converges to the standard chi-square distribution. Furthermore, we obtain the maximum empirical likelihood estimate of the unknown parameter by using the empirical log-likelihood ratio function, and the resulting estimator is shown to be asymptotically normal. Some simulations and an application are conducted to illustrate the proposed method.     

Empirical Likelihood-based Inference for Stationary-ergodicity of the Generalized Random Coefficient Autoregressive Model

In this article, we consider the application of the empirical likelihood method to the generalized random coefficient autoregressive (GRCA) model. When the order of the model is 1, we derive an empirical likelihood ratio test statistic to test the stationary-ergodicity. Some simulation studies are also conducted to investigate the finite sample performances of the proposed test.     

Empirical likelihood inference for partial functional linear model with missing responses

In this paper, we consider the empirical likelihood inferences of the partial functional linear model with missing responses. Two empirical log-likelihood ratios of the parameters of interest are constructed, and the corresponding maximum empirical likelihood estimators of parameters are derived. Under some regularity conditions, we show that the proposed two empirical log-likelihood ratios are asymptotic standard Chi-squared. Thus, the asymptotic results can be used to construct the confidence intervals/regions for the parameters of interest. We also establish the asymptotic distribution theory of corresponding maximum empirical likelihood estimators. A simulation study indicates that the proposed methods are comparable in terms of coverage probabilities and average lengths of confidence intervals. An example of real data is also used to illustrate our proposed methods.     

Numerical Comparison Between Different Empirical Prediction Intervals Under the Fay-Herriot Model

Recently, an empirical best linear unbiased predictor is widely used as a practical approach to small area inference. It is also of interest to construct empirical prediction intervals. However, we do not know which method should be used from among the several existing prediction intervals. In this article, we first obtain an empirical prediction interval by using the residual maximum likelihood method for estimating unknown model variance parameters. Then we compare the later with other intervals with the residual maximum likelihood method. Additionally, some different parametric bootstrap methods for constructing empirical prediction intervals are also compared in a simulation study.     

Weighted empirical likelihood for generalized linear models with longitudinal data

In this paper, we introduce the empirical likelihood (EL) method to longitudinal studies. By considering the dependence within subjects in the auxiliary random vectors, we propose a new weighted empirical likelihood (WEL) inference for generalized linear models with longitudinal data. We show that the weighted empirical likelihood ratio always follows an asymptotically standard chi-squared distribution no matter which working weight matrix that we have chosen, but a well chosen working weight matrix can improve the efficiency of statistical inference. Simulations are conducted to demonstrate the accuracy and efficiency of our proposed WEL method, and a real data set is used to illustrate the proposed method.     

Empirical likelihood inference for general transformation models with right censored data

In this work, we consider empirical likelihood inference for general transformation models with right censored data. The models are a class of flexible semiparametric survival models and include many popular survival models as their special cases. Based on the marginal likelihood function, we define an empirical likelihood ratio statistic. Under some regularity conditions, we show that the empirical likelihood ratio statistic asymptotically follows a standard chi-squared distribution. Through some simulation studies and a real data application, we show that our proposed procedure can work fairly well even for relatively small sample size and high censoring.     

Statistical diagnosis for non-parametric regression models with random right censorship based on the empirical likelihood method

In this paper, we consider statistical diagnostic for non-parametric regression models with right-censored data based on empirical likelihood. First, the primary model is transformed to the non-parametric regression model. Then, based on empirical likelihood methodology, we define some diagnostic statistics. At last, some simulation studies show that our proposed procedure can work fairly well.     

Empirical Likelihood for an Autoregressive Model with Explanatory Variables

In this article, we use the empirical likelihood method to construct the confidence region for parameters in autoregressive model with martingale difference error. It is shown that the empirical log-likelihood ratio at the true parameter converges to the standard chi-square distribution. The simulation results suggest that the empirical likelihood method outperforms the normal approximation based method in terms of coverage probability.     

Empirical Likelihood for First-order Autoregressive Error-in-variable of Models With Validation Data

In this article, we consider the empirical likelihood for the autoregressive error-in-explanatory variable models. With the help of validation, we first develop an empirical likelihood ratio test statistic for the parameters of interest, and prove that its asymptotic distribution is that of a weighted sum of independent standard χ²₁ random variables with unknown weights. Also, we propose an adjusted empirical likelihood and prove that its asymptotic distribution is a standard χ². Furthermore, an empirical likelihood-based confidence region is given. Simulation results indicate that the proposed method works well for practical situations.     

Comparisons of the Unbiased Ridge Estimation to the Other Estimations

In this article, we employ the method of empirical likelihood to construct confidence intervals of conditional density for a 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.     

Empirical likelihood inference for probability density functions under association

In this paper, we consider to apply the empirical likelihood method to a probability density function under an associated sample. It is shown that the empirical likelihood ratio statistic is asymptotically χ²-type distributed under some mild conditions. The result is used to construct empirical likelihood-based confidence intervals on the probability density function.     

Two-sample empirical likelihood method for difference between coefficients in linear regression model

The empirical likelihood method is proposed to construct the confidence regions for the difference in value between coefficients of two-sample linear regression model. Unlike existing empirical likelihood procedures for one-sample linear regression models, as the empirical likelihood ratio function is not concave, the usual maximum empirical likelihood estimation cannot be obtained directly. To overcome this problem, we propose to incorporate a natural and well-explained restriction into likelihood function and obtain a restricted empirical likelihood ratio statistic (RELR). It is shown that RELR has an asymptotic chi-squared distribution. Furthermore, to improve the coverage accuracy of the confidence regions, a Bartlett correction is applied. The effectiveness of the proposed approach is demonstrated by a simulation study.     

Robust empirical likelihood for linear models under median constraints

In this paper, we consider the application of the empirical likelihood for

linear models under median constraints in view of robustness. For two simple median constraints, it is shown that conditions to ensure the consistency of the empirical likelihood confidence regions can be surprisingly relaxed compared with the normal approach under L norm. However, the coverage accuracy of the empirical likelihood confidence regions based on simple median constrains cannot be improved because of the discontinuity of the constraints. Therefore, a smoothed version of median constraint is proposed and a general theory is established to ensure its validity.     

AN EMPIRICAL LIKELIHOOD APPROACH FOR NON‐GAUSSIAN VECTOR STATIONARY PROCESSES AND ITS APPLICATION TO MINIMUM CONTRAST ESTIMATION

We develop the empirical likelihood approach for a class of vector‐valued, not necessarily Gaussian, stationary processes with unknown parameters. In time series analysis, it is known that the Whittle likelihood is one of the most fundamental tools with which to obtain a good estimator of unknown parameters, and that the score functions are asymptotically normal. Motivated by the Whittle likelihood, we apply the empirical likelihood approach to its derivative with respect to unknown parameters. We also consider the empirical likelihood approach to minimum contrast estimation based on a spectral disparity measure, and apply the approach to the derivative of the spectral disparity. This paper provides rigorous proofs on the convergence of our two empirical likelihood ratio statistics to sums of gamma distributions. Because the fitted spectral model may be different from the true spectral structure, the results enable us to construct confidence regions for various important time series parameters without assuming specified spectral structures and the Gaussianity of the process.     

CONFIDENCE INTERVALS FOR THE DIFFERENCE BETWEEN TWO PARTIAL AUCS

As new diagnostic tests are developed and marketed, it is very important to be able to compare the accuracy of a given two continuous‐scale diagnostic tests. An effective method to evaluate the difference between the diagnostic accuracy of two tests is to compare partial areas under the receiver operating characteristic curves (AUCs). In this paper, we review existing parametric methods. Then, we propose a new semiparametric method and a new nonparametric method to investigate the difference between two partial AUCs. For the difference between two partial AUCs under each method, we derive a normal approximation, define an empirical log‐likelihood ratio, and show that the empirical log‐likelihood ratio follows a scaled chi‐square distribution. We construct five confidence intervals for the difference based on normal approximation, bootstrap, and empirical likelihood methods. Finally, extensive simulation studies are conducted to compare the finite‐sample performances of these intervals, and a real example is used as an application of our recommended intervals. The simulation results indicate that the proposed hybrid bootstrap and empirical likelihood intervals outperform other existing intervals in most cases.