Empirical Likelihood Confidence Intervals Research Articles

Empirical Likelihood Confidence Intervals for Lorenz Curve with Length-Biased Data

Empirical Likelihood Confidence Intervals for Lorenz Curve with Length-Biased Data

Empirical likelihood confidence interval for difference-in-differences estimator with panel data

This paper develops an empirical likelihood approach to construct the confidence interval for the average treatment effect on the treated under the difference-in-differences framework. The empirical likelihood function is constructed based on a doubly robust moment function of the parameter of interest. Under some regularity conditions, we show that the proposed method retains the nonparametric Wilks property of empirical likelihood.

Empirical likelihood inference for the mean past lifetime function

In survival analysis and reliability theory, the mean past lifetime (MPL), arises in situations where the mean time elapsed since the failure of a component T, given that it has failed before time t, is of interest. For inference on the MPL function, some procedures have been proposed in the literature for the MPL function's estimator. In this paper, an empirical likelihood (EL) inference procedure of the MPL function is developed. In addition to that, we obtain the adjusted EL and mean EL confidence interval for the MPL function. The EL confidence intervals are compared through extensive simulation studies in terms of coverage probability and the average length of the confidence interval. The simulation studies showed that the proposed EL methods have better coverage probability and shorter average lengths than the normal approximation results. Finally, the proposed methods are illustrated by two real data analyses.

Empirical likelihood confidence interval for sensitivity to the early disease stage.

Disease status can naturally be classified into three or more ordinal stages rather than just being binary stages. Many works have been done for the estimation and inference procedure regarding three ordinal disease stages, which are non-disease, early disease, and full disease stages. The early disease stage can be very important for therapeutic intervention and prevention potentiality. As a diagnostic measure, sensitivity to the early disease stage is often used. In this article, we propose confidence intervals for the sensitivity to early disease stage based on given target specificity for non-disease stage and target sensitivity to full disease stage using both empirical likelihood (EL) and adjusted EL procedures. We compare the performance of the proposed EL confidence intervals with other procedures in our simulation study. The proposed procedures are further applied to Alzheimer's Disease Neuroimaging Initiative data set.

Jackknife empirical likelihood confidence intervals for assessing heterogeneity in meta-analysis of rare binary event data

In meta-analysis, the heterogeneity of effect sizes across component studies is typically described by a variance parameter in a random-effects (Re) model. In the literature, methods for constructing confidence intervals (CIs) for the parameter often assume that study-level effect sizes are normally distributed. However, this assumption might be violated in practice, especially in meta-analysis of rare binary events. We propose to use jackknife empirical likelihood (JEL), a nonparametric approach that uses jackknife pseudo-values, to construct CIs for the heterogeneity parameter. To compute jackknife pseudo-values, we employ a moment-based estimator and consider two commonly used weighing schemes (i.e., equal and inverse variance weights). We prove that with each scheme, the resulting log empirical likelihood ratio follows a chi-square distribution asymptotically. We further examine the performance of the proposed JEL methods and compare them with existing CIs through simulation studies and data examples that focus on data of rare binary events. Our numerical results suggest that the JEL method with equal weights compares favorably to alternatives, especially when (observed) effect sizes are non-normal and the number of component studies is large. Thus, it is worth serious consideration in statistical inference.

Empirical Likelihood Confidence Interval for Difference-in-Differences Estimator with Panel Data 1 a B,†

Empirical Likelihood Confidence Interval for Difference-in-Differences Estimator with Panel Data 1 a B,†

Empirical likelihood for mean difference between two samples with missing data

In this paper, we consider the empirical likelihood confidence intervals for mean difference under the assumption of missing at random (MAR). We prove that the empirical log-likelihood ratio of mean difference converges to a standard chi-squared distribution. We conduct simulation study to compare the proposed empirical likelihood with the normal approximation method in terms of the coverage accuracies and average lengths of confidence intervals. By Monte Carlo simulations, we verify that the proposed method performs well.

New empirical likelihood inference for the mean residual life with length-biased and right-censored data

ABSTRACT The mean residual life (MRL) function for a given random variable T is the expected remaining lifetime of T after a fixed time point t. It is of great interest in survival analysis, reliability, actuarial applications, duration modelling, etc. Liang, Shen, and He [‘Likelihood Ratio Inference for Mean Residual Life of Length-biased Random Variable’, Acta Mathematicae Applicatae Sinica, English Series, 32, 269–282] proposed empirical likelihood (EL) confidence intervals for the MRL based on length-biased right-censored data. However, their -2log(EL ratio) has a scaled chi-squared distribution. To avoid the estimation of the scale parameter in constructing confidence intervals, we propose a new empirical likelihood (NEL) based on i.i.d. representation of Kaplan–Meier weights involved in the estimating equation. We also develop the adjusted new empirical likelihood (ANEL) to improve the coverage probability for small samples. The performance of the NEL and the ANEL compared to the existing EL is demonstrated via simulations: the NEL-based and ANEL-based confidence intervals have better coverage accuracy than the EL-based confidence intervals. Finally, our methods are illustrated with a real data set.

Calibration procedures for linear regression models with multiplicative distortion measurement errors

This paper considers linear regression models when neither the response variable nor the covariates can be directly observed, but are measured with multiplicative distortion measurement errors. To eliminate the effect caused by the distortion, we propose two calibration procedures: the conditional absolute mean calibration and the conditional variance calibration. Both calibration procedures avoid using the nonzero expectation conditions imposed on the variables in the literature. Utilizing these calibrated variables, the least squares estimators are obtained, associated with their asymptotic results. The asymptotic normal confidence intervals and empirical likelihood confidence intervals are also proposed. Simulation studies are conducted to compare the proposed calibration procedures and a real example is analyzed to illustrate our proposed method.

Confidence Intervals of Gini Coefficient Under Unequal Probability Sampling

Abstract We propose an estimator for the Gini coefficient, based on a ratio of means. We show how bootstrap and empirical likelihood can be combined to construct confidence intervals. Our simulation study shows the estimator proposed is usually less biased than customary estimators. The observed coverages of the empirical likelihood confidence interval proposed are also closer to the nominal value.

Nonlinear regression models with profile nonlinear least squares estimation

This paper considers the efficient estimation for a parametric regression model. For parameters estimation, three estimation methods for the parameters are proposed. These estimators are the semi-parametric profile nonlinear least squares estimators, the nonlinear least squares estimators and one-step estimators. We study the asymptotic properties of the proposed estimators, and further discuss their estimation efficiency. The asymptotic normal confidence intervals and empirical likelihood confidence intervals are also proposed for parameters. Simulation studies are conducted to compare the proposed estimation methods.

Multiplicative distortion measurement errors linear models with general moment identifiability condition

ABSTRACTThis paper considers linear regression models when neither the response variable nor the covariates can be directly observed, but are measured with multiplicative distortion measurement errors. The distortion functions for this kind of measurement errors are modelled under a general identifiability condition. For parameter estimation, we propose two calibration procedures: the conditional mean calibration based least squares estimation and the varying coefficient based estimation. The asymptotic normal confidence intervals and empirical likelihood confidence intervals are also proposed. Simulation studies are conducted to compare the proposed calibration procedures and a real example is analysed to illustrate its practical usage.

Influence function-based empirical likelihood for inference of quantile medical costs with censored data.

In this paper, we propose empirical likelihood methods based on influence function and Jackknife techniques to construct confidence intervals for quantile medical costs with censored data. We show that the influence function-based empirical log-likelihood ratio statistic for the quantile medical cost has a standard Chi-square distribution as its asymptotic distribution. Simulation studies are conducted to compare coverage probabilities and interval lengths of the proposed empirical likelihood confidence intervals with the existing normal approximation-based confidence intervals for quantile medical costs. The proposed methods are observed to have better finite-sample performances than existing methods. The new methods are also illustrated through a real example.

Empirical likelihood confidence intervals under imputation for missing survey data from stratified simple random sampling

AbstractMissing observations due to non‐response are commonly encountered in data collected from sample surveys. The focus of this article is on item non‐response which is often handled by filling in (or imputing) missing values using the observed responses (donors). Random imputation (single or fractional) is used within homogeneous imputation classes that are formed on the basis of categorical auxiliary variables observed on all the sampled units. A uniform response rate within classes is assumed, but that rate is allowed to vary across classes. We construct confidence intervals (CIs) for a population parameter that is defined as the solution to a smooth estimating equation with data collected using stratified simple random sampling. The imputation classes are assumed to be formed across strata. Fractional imputation with a fixed number of random draws is used to obtain an imputed estimating function. An empirical likelihood inference method under the fractional imputation is proposed and its asymptotic properties are derived. Two asymptotically correct bootstrap methods are developed for constructing the desired CIs. In a simulation study, the proposed bootstrap methods are shown to outperform traditional bootstrap methods and some non‐bootstrap competitors under various simulation settings. The Canadian Journal of Statistics 47: 281–301; 2019 © 2019 Statistical Society of Canada

Jackknife empirical likelihood-based inference for S-Gini indices

The widely used income inequality measure, Gini index, is extended to form a family of income inequality measures known as Single-Series Gini (S-Gini) indies. In this study, we develop empirical likelihood (EL) and jackknife empirical likelihood (JEL) based inference for the S-Gini indices. We prove that the limiting distribution of both EL and JEL ratio statistics are Chi-square distributions with one degree of freedom. Using the asymptotic distribution we construct EL and JEL based confidence intervals for relative S-Gini indices. We also give bootstrap-t and bootstrap calibrated empirical likelihood confidence intervals for the S-Gini indices. A numerical study is carried out to compare the performances of the proposed asymptotic confidence interval and the bootstrap methods. A test for S-Gini indices based on jackknife empirical likelihood ratio is also proposed. Finally, we illustrate the proposed method using an income data.

Nonparametric interval estimation for the mean of a zero-inflated population

In many statistical analysis, data may consist of excess zero values and the non-zero values are highly positively skewed. Confidence intervals based on a normal approximation for such zero-inflated data may have low coverage probabilities. An empirical likelihood (EL) and adjusted empirical likelihood methods are proposed to construct a non-parametric confidence interval for the mean of zero-inflated population, which are very simple and easy to implement. These EL confidence intervals were compared with existing confidence intervals. Simulation studies are carried out. Finally, the method is implemented in a real data set.

Data driven confidence intervals for diffusion process using double smoothing empirical likelihood

In this paper, we propose an innovative estimation method for the volatility function in diffusion process and construct the empirical likelihood confidence interval for it. Compared with double smoothing local constant estimator, double smoothing local linear estimator proposed in this paper is an inventive estimation method. Moreover, we find that the empirical likelihood confidence interval constructed with the approximate estimating equation is much better than the one based on the estimating equation, since the latter undermines the predictability of the estimator. Accordingly, new algorithm for simulation is proposed. Through Monte Carlo simulation and empirical analysis, both these approaches are superior to traditional asymptotic normality confidence interval in terms of coverage rate, etc.

Empirical likelihood for the bivariate survival function under univariate censoring

The bivariate survival function plays an important role in multivariate survival analysis. Using the idea of influence functions, we develop empirical likelihood confidence intervals for the bivariate survival function in the presence of univariate censoring. It is shown that the empirical log-likelihood ratio has an asymptotic standard chi-squared distribution with one degree of freedom. A comprehensive simulation study shows that the proposed method outperforms both the traditional normal approximation method and the adjusted empirical likelihood method in most cases. The Diabetic Retinopathy Data are analyzed for illustration of the proposed procedure.

Asymptotics of the weighted least squares estimation for AR(1) processes with applications to confidence intervals

For the first-order autoregressive model, we establish the asymptotic theory of the weighted least squares estimations whether the underlying autoregressive process is stationary, unit root, near integrated or even explosive under a weaker moment condition of innovations. The asymptotic limit of this estimator is always normal. It is shown that the empirical log-likelihood ratio at the true parameter converges to the standard chi-square distribution. An empirical likelihood confidence interval is proposed for interval estimations of the autoregressive coefficient. The results improve the corresponding ones of Chan et al. (Econ Theory 28:705–717, 2012). Some simulations are conducted to illustrate the proposed method.

Dimension-reduced empirical likelihood inference for response mean with data missing at random

ABSTRACTTo make efficient inference for mean of a response variable when the data are missing at random and the dimension of covariate is not low, we construct three bias-corrected empirical likelihood (EL) methods in conjunction with dimension-reduced kernel estimation of propensity or/and conditional mean response function. Consistency and asymptotic normality of the maximum dimension-reduced EL estimators are established. We further study the asymptotic properties of the resulting dimension-reduced EL ratio functions and the corresponding EL confidence intervals for the response mean are constructed. The finite-sample performance of the proposed estimators is studied through simulation, and an application to HIV-CD4 data set is also presented.