Empirical Likelihood Ratio Confidence Intervals Research Articles

Empirical likelihood ratio confidence intervals in terms of cumulative hazard function for left-truncated and right-censored data

Based on the approach of Pan and Zhou, we demonstrate that empirical likelihood ratios in terms of cumulative hazard function for left-truncated and right-censored (LTRC) data can be used to form confidence intervals for the parameters that are linear functionals of the cumulative hazard function. Simulation studies indicate that the empirical likelihood ratio based confidence intervals work well in finite samples.

Empirical likelihood inference for non-randomized pretest-posttest studies with missing data

Pretest-posttest studies are commonly used for assessing the effect of a treatment or an intervention. We propose an empirical likelihood based approach to both testing and estimation of the treatment effect in non-randomized pretest-posttest studies where the posttest outcomes are subject to missingness. The proposed empirical likelihood ratio test and the estimation procedure are multiply robust in the sense that multiple working models are allowed for the propensity score of treatment assignment, the missingness probability and the outcome regression, and the validity of the test and the estimation requires only a certain combination of those multiple working models to be correctly specified. An empirical likelihood ratio confidence interval can be constructed for the treatment effect and has better coverage probabilities than confidence intervals based on the Wald statistic. Simulations are conducted to demonstrate the finite-sample performances of the proposed methods.

Comparison of Several Confidence Intervals for Median Residual Lifetime with Left-truncated and Right-censored Data

For left-truncated and right-censored data, the technique proposed by Brookmeyer and Crowley (1982) is extended to construct a point-wise confidence interval for median residual lifetime. This procedure is computationally simpler than the score type confidence interval in Jeong et al. (2008) and empirical likelihood ratio confidence interval in Zhou and Jeong (2011). Further, transformations of the estimator are applied to improve the approximation to the asymptotic distribution for small sample sizes. A simulation study is conducted to investigate the accuracy of these confidence intervals and the implementation of these confidence intervals to two real datasets is illustrated.

Empirical likelihood ratio confidence interval estimation of best linear combinations of biomarkers

A novel smoothed empirical likelihood (EL) approach that incorporates kernel estimation of the area under the receiver operating characteristic curve (AUC) to construct nonparametric confidence intervals of AUC based on the best linear combination (BLC) of biomarkers is proposed. The method has several advantages including the feasibility to use gradient-based techniques for fast computation of BLC coefficients and to employ powerful likelihood methods without specification of underlying data distributions. Simulation results show that the new method performs well even when the distribution of biomarkers is skewed, a situation commonly met in practice. A data set from a clinical experiment related to atherosclerotic coronary heart disease is used to illustrate the efficiency of the proposed method.

Finite-sample properties of the adjusted empirical likelihood

Empirical likelihood-based confidence intervals for the population mean have many interesting properties [Owen, A.B. (1988), ‘Empirical Likelihood Ratio Confidence Intervals for a Single Functional’, Biometrika, 75, 237–249]. Calibrated by χ2 limiting distribution, however, their coverage probabilities are often lower than the nominal when the sample size is small and/or the dimension of the data is high. The application of adjusted empirical likelihood (AEL) is one of the many ways to achieve a more accurate coverage probability. In this paper, we study the finite-sample properties of the AEL. We find that the AEL ratio function decreases when the level of adjustment increases. Thus, the AEL confidence region has higher coverage probabilities when the level of adjustment increases. We also prove that the AEL ratio function increases when the putative population mean moves away from the sample mean. In addition, we show that the AEL confidence region for the population mean is convex. Finally, computer simulations are conducted to further investigate the precision of the coverage probabilities and the sizes of the confidence regions. An application example is also included.

Empirical likelihood ratio confidence intervals for conditional survival probabilities with right censored data

Empirical likelihood ratio confidence intervals for conditional survival probabilities with right censored data

Pseudo–Empirical Likelihood Inference for Multiple Frame Surveys

This article presents a pseudo–empirical likelihood approach to inference for multiple-frame surveys. We establish a unified framework for point and interval estimation of finite population parameters, and show that inferences on the parameters of interest making effective use of different types of auxiliary population information can be conveniently carried out through the constrained maximization of the pseudo–empirical likelihood function. Confidence intervals are constructed using either the asymptotic χ2 distribution of an adjusted pseudo–empirical likelihood ratio statistic or a bootstrap calibration method. Simulation results based on Statistics Canada’s Family Expenditure Survey data show that the proposed methods perform well in finite samples for both point and interval estimation. In particular, a multiplicity-based pseudo–empirical likelihood method is proposed. This method is easily used for multiple-frame surveys with more than two frames and does not require complete frame membership information. The proposed pseudo–empirical likelihood ratio confidence intervals have a clear advantage over the conventional normal approximation–based intervals in estimating population proportions of rare items, a scenario that often motivates the use of multiple-frame surveys. All related computational problems can be handled using existing algorithms for pseudo–empirical likelihood methods with single-frame surveys.

Empirical likelihood confidence intervals for the Gini measure of income inequality

Gini coefficient is among the most popular and widely used measures of income inequality in economic studies, with various extensions and applications in finance and other related areas. This paper studies confidence intervals on the Gini coefficient for simple random samples, using normal approximation, bootstrap percentile, bootstrap-t and the empirical likelihood method. Through both theory and simulation studies it is shown that the intervals based on normal or bootstrap approximation are less satisfactory for samples of small or moderate size than the bootstrap-calibrated empirical likelihood ratio confidence intervals which perform well for all sample sizes. Results for stratified random sampling are also presented.

Bootstrap procedures for the pseudo empirical likelihood method in sample surveys

Pseudo empirical likelihood ratio confidence intervals for finite population parameters are based on asymptotic χ 2 approximation to an adjusted pseudo empirical likelihood ratio statistic, with the adjustment factor related to the design effect. The calculation of the design effect involves variance estimation and hence requires second order inclusion probabilities. It also depends on how auxiliary information is used, and needs to be derived one-at-a-time for different scenarios. This paper presents bootstrap procedures for constructing pseudo empirical likelihood ratio confidence intervals. The proposed method bypasses the need for design effects and is valid under general single-stage unequal probability sampling designs with small sampling fractions. Different scenarios in using auxiliary information are handled by simply including the same type of benchmark constraints with the bootstrap procedures. Simulation results show that the bootstrap calibrated intervals perform very well and have much improved coverage probabilities over the χ 2 -based intervals when the sample sizes are small or moderate.

Weighted empirical likelihood inference for multiple samples

We propose a weighted empirical likelihood approach to inference with multiple samples, including stratified sampling, the estimation of a common mean using several independent and non-homogeneous samples and inference on a particular population using other related samples. The weighting scheme and the basic result are motivated and established under stratified sampling. We show that the proposed method can ideally be applied to the common mean problem and problems with related samples. The proposed weighted approach not only provides a unified framework for inference with multiple samples, including two-sample problems, but also facilitates asymptotic derivations and computational methods. A bootstrap procedure is also proposed in conjunction with the weighted approach to provide better coverage probabilities for the weighted empirical likelihood ratio confidence intervals. Simulation studies show that the weighted empirical likelihood confidence intervals perform better than existing ones.

Smoothed weighted empirical likelihood ratio confidence intervals for quantiles

Thus far, likelihood-based interval estimates for quantiles have not been studied in the literature on interval censored case 2 data and partly interval censored data, and, in this context, the use of smoothing has not been considered for any type of censored data. This article constructs smoothed weighted empirical likelihood ratio confidence intervals (WELRCI) for quantiles in a unified framework for various types of censored data, including right censored data, doubly censored data, interval censored data and partly interval censored data. The fourth order expansion of the weighted empirical log-likelihood ratio is derived and the theoretical coverage accuracy equation for the proposed WELRCI is established, which generally guarantees at least ‘first order’ accuracy. In particular, for right censored data, we show that the coverage accuracy is at least $O(n^{−1/2})$ and our simulation studies show that in comparison with empirical likelihood-based methods, the smoothing used in WELRCI generally provides a shorter confidence interval with comparable coverage accuracy. For interval censored data, it is interesting to find that with an adjusted rate $n^{−1/3}$, the weighted empirical log-likelihood ratio has an asymptotic distribution completely different from that obtained by the empirical likelihood approach and the resulting WELRCI perform favorably in the available comparison simulation studies.

Empirical Likelihood Ratio Confidence Interval for Positively Associated Series

Empirical likelihood is discussed by using the blockwise technique for strongly stationary, positively associated random variables. Our results show that the statistics is asymptotically chi-square distributed and the corresponding confidence interval can be constructed.

Empirical likelihood based inference for the derivative of the nonparametric regression function

We study statistical inference for the derivative of the nonparametric regression function using local linear model based empirical likelihood. We first derive a normal equation for the derivative through the local linear model and use this equation to construct an empirical likelihood for the derivative. We show that the limiting distribution of the empirical likelihood ratio is a scaled χ 1 2 distribution rather than the usual (unscaled) χ 1 2 distribution. We use this limiting distribution to construct pointwise confidence intervals for the derivative. Such empirical likelihood ratio confidence intervals are easier to obtain than the normal approximation based confidence intervals. A small simulation study also suggests that they are more accurate.

EMPIRICAL LIKELIHOOD RATIO CONFIDENCE INTERVAL FOR THE TRIMMED MEAN

ABSTRACT The method of empirical likelihood is extended to make inference for the asymptotic mean of the trimmed mean. In this case, the limiting distribution of the empirical likelihood ratio is not the usual but a scaled . Simulation results indicate that the empirical likelihood ratio confidence interval is more reliable than the normal approximation based interval when the underlying distribution is skewed.

A small sample calibration method for the empirical likelihood ratio

We present a calibration method for improving the coverage accuracy of the empirical likelihood ratio confidence interval for the mean. The method is made possible by a linear transformation invariance property of its coverage level. Simulation results show that, for non-normal distributions, the coverage level of the normal distribution calibrated empirical likelihood ratio confidence interval is comparable to that of the estimated Bartlett-corrected interval. For normal distributions, its coverage level is exact and it is competitive to the t-interval in terms of the variance and expected value of the interval length.

Weighted Empirical Likelihood Ratio Confidence Intervals for the Mean with Censored Data

We propose a procedure to construct the empirical likelihood ratio confidence interval for the mean using a resampling method. This approach leads to the definition of a likelihood function for censored data, called weighted empirical likelihood function. With the second order expansion of the log likelihood ratio, a weighted empirical likelihood ratio confidence interval for the mean is proposed and shown by simulation studies to have comparable coverage accuracy to alternative methods, including the nonparametric bootstrap-t. The procedures proposed here apply in a unified way to different types of censored data, such as right censored data, doubly censored data and interval censored data, and computationally more efficient than the bootstrap-t method. An example of a set of doubly censored breast cancer data is presented with the application of our methods.

Empirical likelihood inference under stratified random sampling using auxiliary population information

SUMMARY The empirical likelihood method under stratified random sampling is used for making inferences on finite population parameters. Our results show that it can lead to efficient estimators by making effective use of auxiliary population information in the form of overall population totals or means ascertained from external sources. Empirical likelihood ratio confidence intervals for the population mean are also studied, and adjusted intervals with asymptotically correct coverage rates are obtained.

Empirical likelihood for partial linear models with fixed designs

Abstract The empirical likelihood method of Owen [Owen, A., 1988. Empirical likelihood ratio confidence intervals for single functional. Biometrika 75, 237–249], is extended to partial linear models with fixed designs in this paper. A nonparametric version of Wilks' theorem is derived. The result is then used to construct confidence regions of the parameter vector in the partial linear models with asymptotically correct coverage probabilities.

Discussion of the Papers by Hinkley and Diciccio and Romano

Discussion of the Papers by Hinkley and Diciccio and Romano

Empirical likelihood ratio confidence intervals for a single functional

SUMMARY The empirical distribution function based on a sample is well known to be the maximum likelihood estimate of the distribution from which the sample was taken. In this paper the likelihood function for distributions is used to define a likelihood ratio function for distributions. It is shown that this empirical likelihood ratio function can be used to construct confidence intervals for the sample mean, for a class of M-estimates that includes quantiles, and for differentiable statistical functionals. The results are nonparametric extensions of Wilks's (1938) theorem for parametric likelihood ratios. The intervals are illustrated on some real data and compared in a simulation to some bootstrap confidence intervals and to intervals based on Student's t statistic. A hybrid method that uses the bootstrap to determine critical values of the likelihood ratio is introduced.